 

Paul Pollack
Remarks on a paper of Ballot and Luca concerning prime divisors of a^{f(n)}1 view print


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


Abstract
Let a be an integer with a > 1. Let f(T) ∈ Q[T] be a nonconstant, integervalued 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^{f(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 a^{f(n)}1 is at most (as x→∞) π(x)/(loglog(x))^{rf+o(1)}, where r_{f} 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))^{1rf}, which we believe is the correct order of magnitude.


Author information
Simon Fraser University, Department of Mathematics, Burnaby, BC Canada V5A 1S6
pollack@math.ubc.ca

