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The continuing search for Wieferich primes

Author(s): Joshua Knauer; Jörg Richstein.
Journal: Math. Comp. 74 (2005), 1559-1563.
MSC (2000): Primary 11A07; Secondary 11-04
Posted: January 19, 2005
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Abstract | References | Similar articles | Additional information

Abstract: A prime $p$ satisfying the congruence

\begin{displaymath}2^{p-1} \equiv 1 \pmod{p^2}\end{displaymath}

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
Email: jknauer@cecm.sfu.ca

Jörg Richstein
Affiliation: Institut für Informatik, Justus-Liebig-Universität, Gieß{}en, Germany
Email: Joerg.Richstein@informatik.uni-giessen.de

DOI: 10.1090/S0025-5718-05-01723-0
PII: S 0025-5718(05)01723-0
Received by editor(s): June 18, 2003
Received by editor(s) in revised form: April 11, 2004
Posted: January 19, 2005
Additional Notes: The second author was supported in part by the Killam Trusts.
Copyright of article: Copyright 2005, American Mathematical Society

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