Prospect Primes Between Two Numbers (Paid)

{
    "query_info": {
        "user_start_input": 350,
        "user_end_input": 1000,
        "calculated_initial_density_value": 0,
        "calculated_final_density_value": 1000000
    },
    "average_densities": {
        "all_primes": "7.869600000000",
        "isolated_primes": "0.000000000000"
    }
}
    

{
    "query_info": {
        "user_start_input": 99999,
        "user_end_input": 999999999,
        "calculated_initial_density_value": 0,
        "calculated_initial_density_value_explanation": "start number (99999) rounded down to the previous million",
        "calculated_final_density_value": 1000000000,
        "calculated_final_density_value_explanation": "end number (999999999) rounded up to the next million"
    },
    "average_densities": {
        "all_primes": "5.085141700000",
        "all_primes_explanation": "average primes density numbers (%) (between 0 and 1 000 000 000)",
        "isolated_primes": "0.001538200000",
        "isolated_primes_explanation": "average isolated primes density numbers (%) (between 0 and 1 000 000 000) [ isolated primes details ==> prime numbers that are more than 100 composite numbers away from each of their neighbours, with an average density of 0.008086097174% ]",
        "palindrome_density": "0.000595900000",
        "palindrome_density_explanation": "average palindrome primes density numbers (%) between 0 and 1 000 000 000); [ palindrome prime number details ==> 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) ]",
        "twin_density": "0.685039900000",
        "twin_density_explanation": "average twin primes density numbers (%) between 0 and 1 000 000 000); [ twin prime number details ==> 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) ]",
        "cousin_density": "0.685034600000",
        "cousin_density_explanation": "average cousin primes density numbers (%) between 0 and 1 000 000 000); [ cousin prime number details ==> 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) ]",
        "sexy_density": "1.158574100000",
        "sexy_density_explanation": "average sexy primes density numbers (%) between 0 and 1 000 000 000); [ sexy prime number details ==> 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) ]",
        "reversible_density": "0.480967000000",
        "reversible_density_explanation": "average reversible primes density numbers (%) between 0 and 1 000 000 000); [ reversible prime number details ==> 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) ]",
        "pandigital_density": "0.000322300000",
        "pandigital_density_explanation": "average pandigital primes density numbers (%) between 0 and 1 000 000 000); [ pandigital prime number details ==> 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/) ]",
        "repunit_density": "0.000000100000",
        "repunit_density_explanation": "average repunit primes density numbers (%) between 0 and 1 000 000 000); [ repunit prime number details ==> repunits primes are positive integers in which every digit is one (examples: 11, 1111111111111111111) (reference: https://primes.utm.edu/glossary/page.php?sort=Repunit) ]",
        "mersenne_density": "0.000000800000",
        "mersenne_density_explanation": "average mersenne primes density numbers (%) between 0 and 1 000 000 000); [ mersenne prime number details ==> 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/) ]",
        "fibonacci_density": "0.000001000000",
        "fibonacci_density_explanation": "average fibonacci primes density numbers (%) between 0 and 1 000 000 000); [ fibonacci prime number details ==> prime numbers that are also Fibonacci numbers (examples: 2, 3, 5, 13, 89, 233, 1597) (reference: https://oeis.org/A005478) ]"
    }
}
    

{
    "error": "please include the api key in your query"
}
    

{
    "error": "start number not specified; please include start number has to be an integer > 2"
}
    

{
    "error": "end number not specified; please include end number has to be an integer < 126 568 967 071"
}
    

{
    "error": "minimum allowed start number has to be > 2"
}
    

{
    "error": "start number has to be an integer > 2"
}
    

{
    "error": "please include end number has to be an integer < 126 568 967 071"
}
    

{
    "error": "maximum allowed start number has to be < 126 566 000 000"
}
    

{
    "error": "maximum allowed end number has to be < 126 568 967 071"
}
    

{
    "error": "start number has to be < end number"
}