Ratio between number of prime numbers less than or equal to an integer and integer can have an approximation {prime number theorem}. Number of primes not exceeding number n is PI(n), whose limit can find the prime numbers: limit of PI(n) / (n / log(n)) = 1, as n goes to infinity.
Number of primes is infinite {Euclid's theorem} {Euclid theorem}.
If p is prime, and a is an integer with no common factor with p, then a^(p - 1) / p has remainder one {Fermat's theorem} {Fermat theorem}.
Positive even integers are sums of two primes {Goldbach's hypothesis} {Goldbach hypothesis} {Goldbach's conjecture}.
Odd numbers can be primes plus two times primes {Levy's conjecture} {Levy conjecture}: p' + 2 * q' = 2*n + 1, where n goes from 0 to infinity.
Algorithms {Shor's algorithm} {Shor algorithm} can find prime factors.
modular
Modular arithmetics have circular sets of numbers. Mathematical operations are periodic.
process
Start with mod. Using any number smaller than the mod, take its first, second, and so on, powers and express result in the mod until number sequence shows a repeating pattern. Distance between repeats is period. Divide period by two and use result as mod exponent. If period divided by two is not even number, start over.
factors
Take the integers one above and one below result. Find largest common divisor of number and two integers to calculate number factors.
From natural-number list, cross out all second numbers except for number two, then cross out all third numbers except for number three, and so on {sieve of Eratosthenes} {Eratosthenes sieve}. What remains are prime numbers.
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0225