New York Journal of Mathematics
Volume 12 (2006) 39-45


Christian Ballot and Florian Luca

Prime factors of af(n)-1 with an irreducible polynomial f(x)

Published: April 12, 2006
Keywords: Prime factors, linear recurrences, Chebotarev Density Theorem
Subject: 11N37, 11B37

In this note, we show that if a is an integer not 0 or ±1 and f(X)∈ Q[X] is an integer valued irreducible polynomial of degree d≧ 2, then the set of primes p dividing af(n)-1 for some positive integer n is of (relative) asymptotic density zero.


This paper was written during a very enjoyable visit by the second author to the Laboratoire Nicolas Oresme of the University of Caen; he wishes to express his thanks to that institution for its hospitality and support. He was also partly supported by grants SEP-CONACYT 46755, PAPIIT IN104505 and a Guggenheim Fellowship.

Author information

Christian Ballot:
Laboratoire de Mathématiques Nicolas Oresme, Université de Caen, BP 5186, 14032 Caen Cedex, France

Florian Luca:
Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P. 58089, Morelia, Michoacán, México