Remarks on a paper of Ballot and Luca concerning prime divisors of a f ( n ) − 1

Let a be an integer with |a| > 1. Let f(T ) ∈ Q[T ] be a nonconstant, integer-valued polynomial with positive leading term, and suppose that there are infinitely many primes p for which f does not possess a root modulo p. Under these hypotheses, Ballot and Luca showed that almost all primes p do not divide any number of the form a−1. More precisely, assuming the Generalized Riemann Hypothesis (GRH), their argument gives that the number of primes p ≤ x which do divide numbers of the form a − 1 is at most (as x→∞) π(x) (log log x)f , where rf is the density of primes p for which the congruence f(n) ≡ 0 (mod p) is insoluble. Under GRH, we improve this upper bound to x(log x)−1−rf , which we believe is the correct order of magnitude.