[FOM] Conjectures on Number Theory.

[Jizhan Hong]

Recently, 2 conjectures on number theory keep coming to my mind. One
is the Regular Prime Conjecture and the other is Artin's primitive
root conjecture.

 I wonder if there are any model theorists tackling these kind of
problems. The common feature of these two problems is that they both
require one to prove that there are infinitely many prime numbers
satisfying some property and infinitely many primes not satisfying it.

I know that Ax-Kochen use ultraproduct and AKE theorem for valued
fields to prove a modified version of Artin's
homogeneous-polynomial-root conjecture about Q_p's (p-adic numbers).
But it looks like their method is only suitable to prove that "for all
but finitely many primes" some property is true.

Penny for your thoughts.

Thanks,
Jizhan Hong