An approximation to the twin prime conjecture and the parity phenomenon

TL;DR

This paper proves that for at least one of the values d=2,4,...16, the number p+d has an odd number of prime factors infinitely often, advancing understanding of prime factorization patterns related to twin primes.

Contribution

It establishes the existence of infinitely many primes p for which p+d has an odd number of prime factors for some d in 2,4,...16, addressing the parity problem.

Findings

Proves the existence of infinitely many primes p with p+d having an odd number of prime factors for some d in 2,4,...16.

Advances understanding of the parity phenomenon in prime factorization.

Provides new insights into the twin prime conjecture and related prime distribution problems.

Abstract

Jing Run Chen proved in 1966 that $p+2$ has at most two prime factors for infinitely many primes $p$. However, due to the parity problem we do not know whether $p+2$ has an odd (or even) number of prime factors infinitely often. In the present work it is proved that $p+d$ has an odd number of prime factors for at least one value of d=2,4,...16.