Mathematika

Table of Contents - July 2012 - Volume 58, Issue 02  

Research Article

ON A CONJECTURE OF ERDŐS

Adam Tyler Felixa1 and M. Ram Murtya2

a1 Max Planck Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany (email: felix@mpim-bonn.mpg.de)

a2 Department of Mathematics & Statistics, Queen’s University, Jeffery Hall, University Avenue, Kingston, Ontario, Canada K7L 3N6 (email: murty@mast.queensu.ca)

Abstract

Let a be an integer different from 0, ±1, or a perfect square. We consider a conjecture of Erdős which states that #{p:ℓ a (p)=r}≪ε r ε for any ε>0, where ℓ a (p) is the order of a modulo p. In particular, we see what this conjecture says about Artin’s primitive root conjecture and compare it to the generalized Riemann hypothesis and the ABC conjecture. We also extend work of Goldfeld related to divisors of p+a and the order of a modulo p.

(Received August 20 2011)

(Online publication February 23 2012)

MSC (2010)

  • 11N13;
  • 11N56;
  • 11N64 (primary)

Footnotes

The first author’s research was supported by an NSERC PGS-D and the second author’s research was supported by an NSERC Discovery Grant.