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ON A CONJECTURE OF ERDŐS

Published online by Cambridge University Press:  23 February 2012

Adam Tyler Felix
Affiliation:
Max Planck Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany (email: [email protected])
M. Ram Murty
Affiliation:
Department of Mathematics & Statistics, Queen’s University, Jeffery Hall, University Avenue, Kingston, Ontario, Canada K7L 3N6 (email: [email protected])
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Abstract

Let a be an integer different from 0, ±1, or a perfect square. We consider a conjecture of Erdős which states that #{p:a(p)=r}≪εrε for any ε>0, where 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.

Type
Research Article
Copyright
Copyright © University College London 2012

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