Relationships and Algorithm in order to achieve the Largest Primes

TL;DR

This paper introduces new theorems and relations that extend beyond Mersenne's theorem, enabling faster access to large prime numbers through flexible formulas and an accompanying algorithm.

Contribution

It presents novel theorems and relations that allow defining formulas for prime numbers in any interval, improving the efficiency of finding large primes.

Findings

Developed extended theorems surpassing Mersenne's theorem

Provided relations enabling formulas for primes in any interval

Proposed an algorithm for finding the largest prime numbers

Abstract

Today, prime numbers attained exceptional situation in the area of numbers theory and cryptography. As we know, the trend for accessing to the largest prime numbers due to using Mersenne theorem, although resulted in vast development of related numbers, however it has reduced the speed of accessing to prime numbers from one to five years. This paper could attain to theorems that are more extended than Mersenne theorem with accelerating the speed of accessing to prime numbers. Since that time, the reason for frequently using Mersenne theorem was that no one could find an efficient formula for accessing to the largest prime numbers. This paper provided some relations for prime numbers that one could define several formulas for attaining prime numbers in any interval; therefore, according to flexibility of these relations, it could be found a new branch in the field of accessing to great prime numbers followed by providing an algorithm at the end of this paper for finding the largest prime numbers.