An elementary conjecture which implies the Goldbach conjecture

TL;DR

The paper proposes a new elementary conjecture involving prime factors of certain integers, which, if true, would imply the Goldbach conjecture, connecting a simple prime factor condition to a famous unsolved problem.

Contribution

It introduces a novel elementary conjecture relating prime factors of specific integers to the Goldbach conjecture, providing a new perspective on this longstanding problem.

Findings

Goldbach's conjecture follows from the proposed conjecture

The conjecture links prime factor properties to the representation of even numbers as sums of two primes

The note offers an elementary approach to a deep number theory problem

Abstract

Let $p_{1}$, ..., $p_{k}$ be the first $k$ odd primes in succession. Let $n$ be an even integer such that $n > p_{k}$. We conjecture that if none of $n - p_{1}$, ..., $n - p_{k}$ are prime, then at least one of them has a prime factor which is greater than or equal to $p_{k}$. In this brief note, we observe that Goldbach's conjecture follows from this conjecture.