A heuristic model from the last post suggests that the “expected” order of the Galois group associated to a weight one modular form of projective type is infinite. And when one tries to solve the inverse Galois problem for central extensions of this group, one is lead to problems concerning the prime divisors of polynomials and their properties modulo 2. But I don’t know how to answer this type of problems! Here is an analogous question that seems a little tricky to me:
Problem: Show that there are infinitely many integers such that all the odd prime divisors of are of the form 5 modulo 8.
To make the problem slightly easier, one can replace (5 modulo 8) by not (1 modulo 2^m) for any fixed m.
Is this an open problem?