The problem of the least prime number in an arithmetic progression and its applications to Goldbach's conjecture

TL;DR

This paper explores the connection between the least prime in an arithmetic progression and Goldbach's conjecture, refining previous results and proposing a generalized problem with an analogous conjecture.

Contribution

It advances the understanding of the least prime in arithmetic progressions and introduces a generalized problem and conjecture related to Goldbach's conjecture.

Findings

Refined the relation between least primes in progressions and Goldbach's conjecture.

Proposed a generalized problem extending the original question.

Established an analogy of Goldbach's conjecture for the generalized problem.

Abstract

The problem of the least prime number in an arithmetic progression is one of the most important topics in Number Theory. In [11], we are the first to study the relations between this problem and Goldbach's conjecture. In this paper, we further consider its applications to Goldbach's conjecture and refine the result in [11]. Moreover, we also try to generalize the problem of the least prime number in an arithmetic progression and give an analogy of Goldbach's conjecture.