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Mathematics of Computation

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The continuing search for Wieferich primes

Authors: Joshua Knauer and Jörg Richstein
Journal: Math. Comp. 74 (2005), 1559-1563
MSC (2000): Primary 11A07; Secondary 11-04
Published electronically: January 19, 2005
MathSciNet review: 2137018
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Abstract: A prime $p$ satisfying the congruence \[ 2^{p-1} \equiv 1 \pmod {p^2}\] is called a Wieferich prime. Although the number of Wieferich primes is believed to be infinite, the only ones that have been discovered so far are $1093$ and $3511$. This paper describes a search for further solutions. The search was conducted via a large scale Internet based computation. The result that there are no new Wieferich primes less than $1.25 \cdot 10^{15}$ is reported.

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Additional Information

Joshua Knauer
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6 Canada

Jörg Richstein
Affiliation: Institut für Informatik, Justus-Liebig-Universität, Gießen, Germany

Received by editor(s): June 18, 2003
Received by editor(s) in revised form: April 11, 2004
Published electronically: January 19, 2005
Additional Notes: The second author was supported in part by the Killam Trusts.
Article copyright: © Copyright 2005 American Mathematical Society