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.
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.