Goldbach Conjecture and the least prime number in an arithmetic progression

TL;DR

This paper explores the connection between the Goldbach Conjecture and the least prime in an arithmetic progression, proposing a weakened form of Goldbach and showing its implications for representing large even numbers as sums of two primes.

Contribution

It introduces a new weakened version of the Goldbach Conjecture and demonstrates its relation to the Chowla Hypothesis, implying the sum of two primes for large even integers.

Findings

A new weakened form of Goldbach Conjecture is proposed.

The weakened Goldbach and Chowla Hypothesis imply all large even integers are sums of two primes.

The study links prime distribution in arithmetic progressions to Goldbach-related conjectures.

Abstract

In this Note, we try to study the relations between the Goldbach Conjecture and the least prime number in an arithmetic progression. We give a new weakened form of the Goldbach Conjecture. We prove that this weakened form and a weakened form of the Chowla Hypothesis imply that every sufficiently large even integer may be written as the sum of two distinct primes. R\'{e}sum\'{e} La conjecture de Goldbach et le plus petit nombre premier dans une progression arithm\'{e}tique Dans ce document, nous essayons d'\'{e}tudier les relations entre la conjecture de Goldbach et le plus petit nombre premier dans une progression arithm\'{e}tique. Nous donnons une nouvelle forme faible de la conjecture de Goldbach. Nous prouvons que cette forme affaiblie et une forme affaiblie de l'hypoth\`{e}se de Chowla impliquent que tout entier pair suffisamment grand peut \^{e}tre \'{e}crit comme une somme de deux nombres premiers distincts.