Get All Primes Between Two Numbers (Paid)
Returns all prime numbers between two designated start and stop numbers, along with their associated data and optional detailed explanations and prime types!
GET get-all-primes-between-two-numbers (paid)
http://api.prime-numbers.io/get-all-primes-between-two-numbers.php?key=YOUR_API_KEY&include_explanations=<boolean>&include_prime_types_list=<boolean>&language=<string>
AUTHORIZATION API Key
This request is using the API Key from Prime Numbers API collection in thePime Numbers API environment
PARAMS
| key |
YOUR_API_KEY
(Required) your API key |
| include_explanations |
<boolean>
includes the full explanations for each item if true (default is false) |
| include_prime_types_list |
<boolean>
includes the full prime types list for each item if true (default is false) |
| language |
<string>
show the output translated into that language (it can be english, mandarin, hindi, spanish, french, german, italian, japanese, russian) (default is english) |
Success (with start/stop numbers)
Example Request
Example Response
200 OK
[
{
"random_prime_number_value": 353,
"base_conversions": {
"binary_value": "101100001",
"senary_value": "1345",
"hexa_value": "161"
},
"previous_prime_gap": 4,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 19 computations in 0.00078284559283566 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 359,
"base_conversions": {
"binary_value": "101100111",
"senary_value": "1355",
"hexa_value": "167"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 19 computations in 0.00078947063839568 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 361,
"base_conversions": {
"binary_value": "101101001",
"senary_value": "1401",
"hexa_value": "169"
},
"previous_prime_gap": 2,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-09-02 00:24:57: server mac-server processed 19 computations in 0.00079166666666667 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 367,
"base_conversions": {
"binary_value": "101101111",
"senary_value": "1411",
"hexa_value": "16f"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 19 computations in 0.00079821850252783 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 373,
"base_conversions": {
"binary_value": "101110101",
"senary_value": "1421",
"hexa_value": "175"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 19 computations in 0.00080471699649283 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 379,
"base_conversions": {
"binary_value": "101111011",
"senary_value": "1431",
"hexa_value": "17b"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 19 computations in 0.00081116343058049 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 383,
"base_conversions": {
"binary_value": "101111111",
"senary_value": "1435",
"hexa_value": "17f"
},
"previous_prime_gap": 4,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 20 computations in 0.00081543274128254 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 389,
"base_conversions": {
"binary_value": "110000101",
"senary_value": "1445",
"hexa_value": "185"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 20 computations in 0.00082179512180483 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 397,
"base_conversions": {
"binary_value": "110001101",
"senary_value": "1501",
"hexa_value": "18d"
},
"previous_prime_gap": 8,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 20 computations in 0.00083020245188214 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 401,
"base_conversions": {
"binary_value": "110010001",
"senary_value": "1505",
"hexa_value": "191"
},
"previous_prime_gap": 4,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 20 computations in 0.00083437434977087 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 409,
"base_conversions": {
"binary_value": "110011001",
"senary_value": "1521",
"hexa_value": "199"
},
"previous_prime_gap": 8,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 20 computations in 0.00084265618400653 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 419,
"base_conversions": {
"binary_value": "110100011",
"senary_value": "1535",
"hexa_value": "1a3"
},
"previous_prime_gap": 10,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 20 computations in 0.00085289539543578 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 421,
"base_conversions": {
"binary_value": "110100101",
"senary_value": "1541",
"hexa_value": "1a5"
},
"previous_prime_gap": 2,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 21 computations in 0.00085492852202847 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 431,
"base_conversions": {
"binary_value": "110101111",
"senary_value": "1555",
"hexa_value": "1af"
},
"previous_prime_gap": 10,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 21 computations in 0.00086502247883445 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 433,
"base_conversions": {
"binary_value": "110110001",
"senary_value": "2001",
"hexa_value": "1b1"
},
"previous_prime_gap": 2,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 21 computations in 0.00086702716861187 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 439,
"base_conversions": {
"binary_value": "110110111",
"senary_value": "2011",
"hexa_value": "1b7"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 21 computations in 0.00087301361832321 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 443,
"base_conversions": {
"binary_value": "110111011",
"senary_value": "2015",
"hexa_value": "1bb"
},
"previous_prime_gap": 4,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 21 computations in 0.00087698188249372 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 449,
"base_conversions": {
"binary_value": "111000001",
"senary_value": "2025",
"hexa_value": "1c1"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 21 computations in 0.00088290083751738 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 457,
"base_conversions": {
"binary_value": "111001001",
"senary_value": "2041",
"hexa_value": "1c9"
},
"previous_prime_gap": 8,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 21 computations in 0.00089073159693466 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 461,
"base_conversions": {
"binary_value": "111001101",
"senary_value": "2045",
"hexa_value": "1cd"
},
"previous_prime_gap": 4,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 21 computations in 0.000894621273066 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 463,
"base_conversions": {
"binary_value": "111001111",
"senary_value": "2051",
"hexa_value": "1cf"
},
"previous_prime_gap": 2,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 22 computations in 0.00089655978297292 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 467,
"base_conversions": {
"binary_value": "111010011",
"senary_value": "2055",
"hexa_value": "1d3"
},
"previous_prime_gap": 4,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 22 computations in 0.00090042428270726 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 479,
"base_conversions": {
"binary_value": "111011111",
"senary_value": "2115",
"hexa_value": "1df"
},
"previous_prime_gap": 12,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 22 computations in 0.00091191952617664 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 487,
"base_conversions": {
"binary_value": "111100111",
"senary_value": "2131",
"hexa_value": "1e7"
},
"previous_prime_gap": 8,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 22 computations in 0.00091950318711308 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 491,
"base_conversions": {
"binary_value": "111101011",
"senary_value": "2135",
"hexa_value": "1eb"
},
"previous_prime_gap": 4,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 22 computations in 0.00092327165859001 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 499,
"base_conversions": {
"binary_value": "111110011",
"senary_value": "2151",
"hexa_value": "1f3"
},
"previous_prime_gap": 8,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 22 computations in 0.00093076282932036 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 503,
"base_conversions": {
"binary_value": "111110111",
"senary_value": "2155",
"hexa_value": "1f7"
},
"previous_prime_gap": 4,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 22 computations in 0.00093448589550024 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 509,
"base_conversions": {
"binary_value": "111111101",
"senary_value": "2205",
"hexa_value": "1fd"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 23 computations in 0.00094004284772321 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 521,
"base_conversions": {
"binary_value": "1000001001",
"senary_value": "2225",
"hexa_value": "209"
},
"previous_prime_gap": 12,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 23 computations in 0.00095105935087611 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 523,
"base_conversions": {
"binary_value": "1000001011",
"senary_value": "2231",
"hexa_value": "20b"
},
"previous_prime_gap": 2,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 23 computations in 0.00095288305216911 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 529,
"base_conversions": {
"binary_value": "1000010001",
"senary_value": "2241",
"hexa_value": "211"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-09-02 00:24:57: server mac-server processed 23 computations in 0.00095833333333333 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 541,
"base_conversions": {
"binary_value": "1000011101",
"senary_value": "2301",
"hexa_value": "21d"
},
"previous_prime_gap": 12,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 23 computations in 0.00096914194580108 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 547,
"base_conversions": {
"binary_value": "1000100011",
"senary_value": "2311",
"hexa_value": "223"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 23 computations in 0.00097450129696054 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 557,
"base_conversions": {
"binary_value": "1000101101",
"senary_value": "2325",
"hexa_value": "22d"
},
"previous_prime_gap": 10,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 24 computations in 0.00098336864343383 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 563,
"base_conversions": {
"binary_value": "1000110011",
"senary_value": "2335",
"hexa_value": "233"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 24 computations in 0.00098865087647539 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 569,
"base_conversions": {
"binary_value": "1000111001",
"senary_value": "2345",
"hexa_value": "239"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 24 computations in 0.00099390503682305 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 571,
"base_conversions": {
"binary_value": "1000111011",
"senary_value": "2351",
"hexa_value": "23b"
},
"previous_prime_gap": 2,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 24 computations in 0.0009956502621YOUR_API_KEY8 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 577,
"base_conversions": {
"binary_value": "1001000001",
"senary_value": "2401",
"hexa_value": "241"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 24 computations in 0.001000867679122 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 587,
"base_conversions": {
"binary_value": "1001001011",
"senary_value": "2415",
"hexa_value": "24b"
},
"previous_prime_gap": 10,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 24 computations in 0.0010095034532988 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 593,
"base_conversions": {
"binary_value": "1001010001",
"senary_value": "2425",
"hexa_value": "251"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 24 computations in 0.0010146496384905 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 599,
"base_conversions": {
"binary_value": "1001010111",
"senary_value": "2435",
"hexa_value": "257"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 24 computations in 0.00101976985421 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 601,
"base_conversions": {
"binary_value": "1001011001",
"senary_value": "2441",
"hexa_value": "259"
},
"previous_prime_gap": 2,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
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"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 30 computations in 0.00YOUR_API_KEY67351733047 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 883,
"base_conversions": {
"binary_value": "1101110011",
"senary_value": "4031",
"hexa_value": "373"
},
"previous_prime_gap": 2,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 30 computations in 0.00YOUR_API_KEY81381631753 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 887,
"base_conversions": {
"binary_value": "1101110111",
"senary_value": "4035",
"hexa_value": "377"
},
"previous_prime_gap": 4,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 30 computations in 0.0012409393843196 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 899,
"base_conversions": {
"binary_value": "1110000011",
"senary_value": "4055",
"hexa_value": "383"
},
"previous_prime_gap": 12,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-09-02 00:24:58: server mac-server processed 30 computations in 0.0012493053625471 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 907,
"base_conversions": {
"binary_value": "1110001011",
"senary_value": "4111",
"hexa_value": "38b"
},
"previous_prime_gap": 8,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 30 computations in 0.0012548516955313 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 911,
"base_conversions": {
"binary_value": "1110001111",
"senary_value": "4115",
"hexa_value": "38f"
},
"previous_prime_gap": 4,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 30 computations in 0.0012576156893989 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 919,
"base_conversions": {
"binary_value": "1110010111",
"senary_value": "4131",
"hexa_value": "397"
},
"previous_prime_gap": 8,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 30 computations in 0.001263125532602 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 929,
"base_conversions": {
"binary_value": "1110100001",
"senary_value": "4145",
"hexa_value": "3a1"
},
"previous_prime_gap": 10,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 30 computations in 0.0012699792211773 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 937,
"base_conversions": {
"binary_value": "1110101001",
"senary_value": "4201",
"hexa_value": "3a9"
},
"previous_prime_gap": 8,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 31 computations in 0.0012754356554178 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 941,
"base_conversions": {
"binary_value": "1110101101",
"senary_value": "4205",
"hexa_value": "3ad"
},
"previous_prime_gap": 4,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 31 computations in 0.0012781551375148 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 947,
"base_conversions": {
"binary_value": "1110110011",
"senary_value": "4215",
"hexa_value": "3b3"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 31 computations in 0.0012822235461191 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 953,
"base_conversions": {
"binary_value": "1110111001",
"senary_value": "4225",
"hexa_value": "3b9"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 31 computations in 0.0012862790867028 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 961,
"base_conversions": {
"binary_value": "1111000001",
"senary_value": "4241",
"hexa_value": "3c1"
},
"previous_prime_gap": 8,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-09-02 00:24:58: server mac-server processed 31 computations in 0.0012916666666667 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 967,
"base_conversions": {
"binary_value": "1111000111",
"senary_value": "4251",
"hexa_value": "3c7"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 31 computations in 0.0012956926504555 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 971,
"base_conversions": {
"binary_value": "1111001011",
"senary_value": "4255",
"hexa_value": "3cb"
},
"previous_prime_gap": 4,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 31 computations in 0.0012983697042402 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 977,
"base_conversions": {
"binary_value": "1111010001",
"senary_value": "4305",
"hexa_value": "3d1"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 31 computations in 0.0013023749673406 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 983,
"base_conversions": {
"binary_value": "1111010111",
"senary_value": "4315",
"hexa_value": "3d7"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 31 computations in 0.0013063679505492 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 991,
"base_conversions": {
"binary_value": "1111011111",
"senary_value": "4331",
"hexa_value": "3df"
},
"previous_prime_gap": 8,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 31 computations in 0.0013116730198914 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
},
{
"random_prime_number_value": 997,
"base_conversions": {
"binary_value": "1111100101",
"senary_value": "4341",
"hexa_value": "3e5"
},
"previous_prime_gap": 6,
"prime_density": "7.86960000",
"isolated_primes": {
"is_isolated_prime": "false",
"isolated_prime_density": "7.86960000"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 32 computations in 0.0013156377836539 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs"
}
]
Success (with start/stop numbers, explanations, and prime_types)
Example Request
Example Response
200 OK
[
{
"random_prime_number_value": 71,
"base_conversions": {
"binary_value": "1000111",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "155",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "47",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 4,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 73,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 67,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "true",
"reversible_emirp_value": 17,
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 8 computations in 0.00035108957388235 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 73,
"base_conversions": {
"binary_value": "1001001",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "201",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "49",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 2,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 71,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "false",
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 79,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "true",
"reversible_emirp_value": 37,
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 9 computations in 0.0003560001560549 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 79,
"base_conversions": {
"binary_value": "1001111",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "211",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "4f",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 6,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "false",
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 83,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 73,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "true",
"reversible_emirp_value": 97,
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 9 computations in 0.00037034143405482 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 83,
"base_conversions": {
"binary_value": "1010011",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "215",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "53",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 4,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "false",
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 79,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 89,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 9 computations in 0.00037960139913101 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 89,
"base_conversions": {
"binary_value": "1011001",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "225",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "59",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 6,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "false",
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "false",
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 83,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "true",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 9 computations in 0.00039308254716903 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 97,
"base_conversions": {
"binary_value": "1100001",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "241",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "61",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 8,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "false",
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 101,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "true",
"reversible_emirp_value": 79,
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 10 computations in 0.00041036907507484 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 101,
"base_conversions": {
"binary_value": "1100101",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "245",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "65",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 4,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "true",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 103,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 97,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 10 computations in 0.0004187448175467 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 103,
"base_conversions": {
"binary_value": "1100111",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "251",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "67",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 2,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 101,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 107,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 10 computations in 0.00042287048187884 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 107,
"base_conversions": {
"binary_value": "1101011",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "255",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "6b",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 4,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 109,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 103,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "true",
"reversible_emirp_value": 701,
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 10 computations in 0.00043100335136619 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 109,
"base_conversions": {
"binary_value": "1101101",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "301",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "6d",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 2,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 107,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 113,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 10 computations in 0.00043501277120461 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 113,
"base_conversions": {
"binary_value": "1110001",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "305",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "71",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 4,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "false",
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 109,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "true",
"reversible_emirp_value": 311,
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 11 computations in 0.00044292274219728 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 121,
"base_conversions": {
"binary_value": "1111001",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "321",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "79",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 8,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "true",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "false",
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "false",
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 127,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-09-02 00:24:57: server mac-server processed 11 computations in 0.00045833333333333 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 127,
"base_conversions": {
"binary_value": "1111111",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "331",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "7f",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 6,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "false",
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 131,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 121,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "true",
"power_of_two_value": 7,
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 11 computations in 0.00046955948623269 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 131,
"base_conversions": {
"binary_value": "10000011",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "335",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "83",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 4,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "true",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "false",
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 127,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 137,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 11 computations in 0.00047689679759415 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 137,
"base_conversions": {
"binary_value": "10001001",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "345",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "89",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 6,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 139,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "false",
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 131,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 12 computations in 0.00048769582961332 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 139,
"base_conversions": {
"binary_value": "10001011",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "351",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "8b",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 2,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 137,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 143,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 12 computations in 0.00049124275510632 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 143,
"base_conversions": {
"binary_value": "10001111",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "355",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "8f",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 4,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "false",
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 139,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 149,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-09-02 00:24:57: server mac-server processed 12 computations in 0.00049826086429589 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 149,
"base_conversions": {
"binary_value": "10010101",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "405",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "95",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 6,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 151,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "false",
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 143,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "true",
"reversible_emirp_value": 941,
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 12 computations in 0.0005086064839889 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 151,
"base_conversions": {
"binary_value": "10010111",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "411",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "97",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 2,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "true",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 149,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "false",
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 157,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 12 computations in 0.00051200857197685 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 157,
"base_conversions": {
"binary_value": "10011101",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "421",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "9d",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 6,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "false",
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "false",
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 163,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "true",
"reversible_emirp_value": 751,
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 13 computations in 0.00052208183692257 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 163,
"base_conversions": {
"binary_value": "10100011",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "431",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "a3",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 6,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "false",
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 167,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 157,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 13 computations in 0.00053196438895015 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 167,
"base_conversions": {
"binary_value": "10100111",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "435",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "a7",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 4,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 169,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 163,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "true",
"reversible_emirp_value": 761,
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 13 computations in 0.000538451999305 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 169,
"base_conversions": {
"binary_value": "10101001",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "441",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "a9",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 2,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 167,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 173,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-09-02 00:24:57: server mac-server processed 13 computations in 0.00054166666666667 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 173,
"base_conversions": {
"binary_value": "10101101",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "445",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "ad",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 4,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "false",
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 169,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 179,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 13 computations in 0.00054803943491525 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 179,
"base_conversions": {
"binary_value": "10110011",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "455",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "b3",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 6,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 181,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "false",
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "true",
"sexy_value": 173,
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "true",
"reversible_emirp_value": 971,
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 13 computations in 0.00055746200667749 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 181,
"base_conversions": {
"binary_value": "10110101",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "501",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "b5",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 2,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "true",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 179,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "false",
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 13 computations in 0.00056056766862807 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 191,
"base_conversions": {
"binary_value": "10111111",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "515",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "bf",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 10,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "true",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 193,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "false",
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 14 computations in 0.00057584479004522 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 193,
"base_conversions": {
"binary_value": "11000001",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "521",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "c1",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 2,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 191,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 197,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 14 computations in 0.00057885183289374 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 197,
"base_conversions": {
"binary_value": "11000101",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "525",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "c5",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 4,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 199,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "true",
"cousin_value": 193,
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "false",
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 14 computations in 0.00058481953531742 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
},
{
"random_prime_number_value": 199,
"base_conversions": {
"binary_value": "11000111",
"binary_value_explanation": "prime number base-2 (binary value), useful for cryptography and cryptocurrency",
"senary_value": "531",
"senary_value_explanation": "prime number base-6 (senary value), useful for mathematical research",
"hexa_value": "c7",
"hexa_value_explanation": "prime number base-16 (hexa value), useful for cryptography and cryptocurrency"
},
"previous_prime_gap": 2,
"previous_prime_gap_explanation": "how many successive prime and composite numbers are between this prime and the previous one",
"prime_density": "7.86960000",
"prime_density_explanation": "how many prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"isolated_primes": {
"is_isolated_prime": "false",
"is_isolated_prime_explanation": "prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174%",
"isolated_prime_density": "7.86960000",
"isolated_prime_density_explanation": "how many chances (%) to randomly find an isolated prime number in this million composite numbers (between 0 and 1 000 000)"
},
"prime_types": {
"is_palindrome": "false",
"palindrome_explanation": "number that is simultaneously palindromic (which reads the same backwards as forwards) and prime (examples: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929) (reference: https://en.wikipedia.org/wiki/Palindromic_prime)",
"palindrome_percentage": "0.01190000",
"palindrome_density_explanation": "how many palindrome prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_twin": "true",
"twin_value": 197,
"twin_explanation": "primes that are no more than 2 composite numbers from each other (examples: (3, 5), (5, 7), (11, 13), (17, 19)) (reference: https://en.wikipedia.org/wiki/Twin_prime)",
"twin_percentage": "1.64450000",
"twin_density_explanation": "how many twin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_cousin": "false",
"cousin_explanation": "primes that are no more than 4 composite numbers from each other (examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47)) (reference: https://en.wikipedia.org/wiki/Cousin_prime)",
"cousin_percentage": "1.63800000",
"cousin_density_explanation": "how many cousin prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_sexy": "false",
"sexy_explanation": "primes that are no more than 6 composite numbers from each other (examples: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29)) (reference: https://en.wikipedia.org/wiki/Sexy_prime)",
"sexy_percentage": "2.52870000",
"sexy_density_explanation": "how many sexy prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_reversible": "true",
"reversible_emirp_value": 991,
"reversible_explanation": "primes that become a different prime when their decimal digits are reversed. The name emirp is obtained by reversing the word prime (examples: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157) (reference: https://en.wikipedia.org/wiki/Emirp)",
"reversible_percentage": "1.12150000",
"reversible_density_explanation": "how many reversible prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_pandigital": "false",
"pandigital_explanation": "pandigital prime in a base has at least one instance of each base digit. (examples: 2143 (base 4), 7654321 (base 7)) (reference: https://www.xarg.org/puzzle/project-euler/problem-41/)",
"pandigital_percentage": "0.00210000",
"pandigital_density_explanation": "how many pandigital prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_repunit": "false",
"repunit_explanation": "repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit)",
"repunit_percentage": "0.00010000",
"repunit_density_explanation": "how many repunit prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_mersenne": "false",
"mersenne_explanation": "mersenne prime is a prime number that is of the form 2n - 1 (one less than a power of two) for some integer n. They are named after Marin Mersenne (1588-1648), a French monk who studied them in his Cogitata Physica-Mathematica (1644) (examples: 3 (2^2 - 1), 7 (2^3 - 1), 31 (2^5 - 1) ) (reference: https://www.mersenne.org/)",
"mersenne_percentage": "0.00080000",
"mersenne_density_explanation": "how many mersenne prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)",
"is_fibonacci": "false",
"fibonacci_explanation": "prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478)",
"fibonacci_percentage": "0.00090000",
"fibonacci_density_explanation": "how many fibonacci prime numbers (%) can be found in this million composite numbers (between 0 and 1 000 000)"
},
"birth_certificate": "2018-06-16 22:01:25: server mac-server processed 14 computations in 0.00058778066581941 micro-seconds using 2 x 3 GHz Quad-Core Intel Xeon CPUs",
"birth_certificate_explanation": "how many computations, how much time and what computer power was used to find this prime number"
}
]
Error (no key)
Example Request
Example Response
403 Forbidden
{
"error": "please include the api key in your query"
}
Error (no start number)
Example Request
Example Response
404 Not Found
{
"error": "start number not specified; start number has to be an integer > 2"
}
Error (no end number)
Example Request
Example Response
404 Not Found
{
"error": "end number not specified; please include end number has to be an integer < 126 568 967 071"
}
Error (start number too small)
Example Request
Example Response
404 Not Found
{
"error": "minimum allowed start number has to be > 2"
}
Error (alpha start number)
Example Request
Example Response
404 Not Found
{
"error": "start number has to be an integer > 2"
}
Error (alpha end number)
Example Request
Example Response
404 Not Found
{
"error": "please include end number < 126 568 967 071"
}
Error (start number more than maximum)
Example Request
Example Response
404 Not Found
{
"error": "start number has to be < end number"
}
Error (end number more than maximum)
Example Request
Example Response
404 Not Found
{
"error": "maximum allowed end number has to be < 126 568 967 071"
}
Error (no results)
Example Request
Example Response
404 Not Found
{
"error": "no numbers found"
}
Error (end number smaller than the start number)
Example Request
Example Response
404 Not Found
{
"error": "start number has to be < end number"
}