Refined Goldbach conjectures with primes in progressions

TL;DR

This paper proposes refined versions of Goldbach's conjecture involving primes in specific arithmetic progressions, supported by heuristic and numerical evidence, and discusses the expected size of exceptional sets where the conjecture might not hold.

Contribution

It introduces new conjectures on Goldbach's problem with primes in progressions and analyzes the size of potential exceptions based on heuristic and numerical data.

Findings

Conjecture that even numbers >4 are sums of two primes with one prime ≡ 3 mod 4.

Generalization to primes in arithmetic progressions with finite exceptional sets.

Predictions on the growth of exceptional sets where the conjecture may fail.

Abstract

We formulate some refinements of Goldbach's conjectures based on heuristic arguments and numerical data. For instance, any even number greater than 4 is conjectured to be a sum of two primes with one prime being 3 mod 4. In general, for fixed $m$ and $a, b$ coprime to $m$, any positive even $n \equiv a + b \bmod m$ outside of a finite exceptional set is expected to be a sum of two primes $p$ and $q$ with $p \equiv a \bmod m$, $q \equiv b \bmod m$. We make conjectures about the growth of these exceptional sets.