New York Journal of Mathematics
Volume 17 (2011) 553-567


Paul Pollack

Remarks on a paper of Ballot and Luca concerning prime divisors of af(n)-1

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Published: August 17, 2011
Keywords: Prime factors, Chebotarev density theorem, orders modulo p
Subject: Primary: 11N37, Secondary: 11B83

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 af(n)-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 af(n)-1 is at most (as x→∞)
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.

Author information

Simon Fraser University, Department of Mathematics, Burnaby, BC Canada V5A 1S6