The Chebotarev density theorem is often useful in answering questions involving primes and integer polynomials. After a brief recollection of diverse problems that can be tackled in this manner, we answer some specific questions such as: `Given integers a,b and relatively prime positive integers c,d, what is the proportion of primes that are in the arithmetic progression c modulo d and divide a^n+b^n for some n?’

The method of proof originally arose in the study of Artin’s primitive roots conjecture. We mention also some work in progress towards determining (given integers a,b>1) the density of primes p for which a is a power of b modulo p.