Artin’s conjecture and the ternary Goldbach problem

Fix 3  non-square integers a_1,a_2,a_3, none of which being -1.
I will talk about the problem of representating a large odd integer as a sum of 3 primes p_1,p_2,p_3
such that every p_i has a_i as a primitive root. 

Solutions of Diophantine equations with all variables being primes with prescribed primitive roots have not been studied before.

We use the circle method and Hooley's approach for the Artin conjecture
to prove an asymptotic for the number of representations as the odd integer tends to infinity.
The leading constant is connected to work on Artin's constant in various settings and I will focus
on explaining a model that we can verify under GRH.

This is joint work with Peter Koymans (Leiden) and Christopher Frei (Manchester).