Infinitely many pairs of primes $p$ and $p+2$

TL;DR

The paper introduces a sieve-based method to prove the existence of infinitely many twin primes, extending classical results by employing multiple sieves and special conditions.

Contribution

It presents a novel sieve system approach that proves infinitely many twin primes exist without restrictions, advancing prime number theory.

Findings

Proves infinitely many twin primes exist under certain conditions.

Develops a new sieve method for prime pair analysis.

Establishes twin prime infinity without restrictions.

Abstract

We take the pre-sieved set to be all natural numbers $N=\{1,2,3,\dots\}$ with a sieve system:single sieve,double sieve,.... With single sieve, i.e. , remove out the multiple of a prime, we derive all the primes. With double sieve, i.e. , remove out the multiple and the multiple of a prime and $-2$ simultaneously, we get all the prime twins and prove that infinitely many prime twins exist under suitable conditions. Finally, with special 4 sieve, we prove that infinitely many prime twins exist without any restriction.