Refined Goldbach Conjectures with Primes in Progressions

Abstract

We consider binary Goldbach-type conjectures with primes in arithmetic progressions. Namely, we make conjectures about the size of exceptional sets based on heuristic arguments and numerical data. As a specific example, any even number greater than 4 is conjectured to be a sum of two primes with one prime being 3 mod 4.

Keywords:

• Goldbach conjecture
• primes in arithmetic progressions
• exceptional set

Acknowledgments

We thank Claus Bauer, Harald Helfgott and the anonymous referee for some helpful comments.

Declaration of interest

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by a grant from the Simons Foundation/SFARI (512927, KM).