Refined Goldbach conjectures with primes in progressions

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 $n\equiv a+b\mathrm{mod}m$ outside of a finite exceptional set is expected to be a sum of two primes $p$ and $q$ with $p\equiv a\mathrm{mod}m$, $q\equiv b\mathrm{mod}m$. We make conjectures about the growth of these exceptional sets.

arXiv comment10 pages