On a problem of Narkiewicz.
We prove an upper bound for the number of primes p ≤ x in an arithmetic progression 1 (mod Q) that are exceptional in the sense that has no generator in the interval [1,B]. As a consequence we prove that if with a sufficiently large absolute constant c, then there exists a prime q dividing Q such that for some positive integer b ≤ B. Moreover we estimate the number of such q’s under suitable conditions.
In finite Galois extensions of with pairwise coprime discriminants the integral and the prime divisors subject to the condition are equidistributed in the sense of E. Hecke.