On Selberg's approximation to the twin prime problem

TL;DR

This paper refines Selberg's approximation method for the twin prime problem, reducing the upper bound on the sum of prime factors for numbers in twin prime pairs from 14 to approximately 12.59.

Contribution

It improves Selberg's classical bound on prime factors in twin primes by refining the approximation parameter using a suggested enhancement.

Findings

Reduced the bound from 14 to about 12.59

Established the existence of infinitely many twin primes with bounded prime factors

Enhanced the understanding of prime factorization in twin primes

Abstract

In his Classical approximation to the Twin prime problem, Selberg proved that for $x$ sufficiently large, there is an $n \in (x,2x)$ such that $2^{\Omega(n)}+2^{\Omega(n+2)} \leq \lambda$ with $\lambda=14$, where $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity. This enabled him to show that for infinitely many $n$, $n(n+2)$ has atmost $5$ prime factors, with one having atmost $2$ and the other having atmost $3$ prime factors. By adopting Selberg's approach and using a refinement suggested by Selberg, we improve this value of $\lambda$ to about $\lambda=12.59$.