On The Infinitude of the Twin Primes

TL;DR

This paper introduces a new framework for the Twin Prime Conjecture using $6x \u00b1 1$ representations, reformulating it as an existence problem of certain generators and applying sieve methods to analyze their distribution.

Contribution

The paper proposes a novel approach to the Twin Prime Conjecture based on $6x \u00b1 1$ representations and structured sieving techniques, highlighting the parity problem as a key obstacle.

Findings

Partition of natural numbers into structured intervals $\u0107A_n$

Application of sieve to estimate candidate generators

Identification of the parity problem as a major challenge

Abstract

We present a novel approach to the Twin Prime Conjecture, basing on the $6x \pm 1$ representation of primes. By defining so-called twin prime generators $x \in \N$, for which both $6x - 1$ and $6x + 1$ are prime, we reformulate the conjecture into the existence problem of such $x$. Using admissible residue classes modulo products of small primes and an adapted Selberg sieve, we partition the natural numbers into structured intervals $\mc{A}_n$, where the maximal possible prime divisor of $6x \pm 1$ is fixed. Within each $\mc{A}_n$, we apply the sieve to estimate the number of generator candidates that escape all local obstructions. Due to the \emph{parity problem} we cannot solve the problem with a Selberg sieve. It requires other sieves or methods. The author is searching for them and invites all interested people to help.