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Solutions of the congruence $a^{p-1} \equiv 1 \pmod {p^r}$

Authors: Wilfrid Keller and Jörg Richstein
Journal: Math. Comp. 74 (2005), 927-936
MSC (2000): Primary 11A07; Secondary 11D61, 11--04
Published electronically: June 8, 2004
MathSciNet review: 2114655
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Abstract: To supplement existing data, solutions of $a^{p-1} \equiv 1 \pmod {p^2}$ are tabulated for primes $a, p$with $100 < a < 1000$ and $10^4 < p < 10^{11}$. For $a < 100$, five new solutions $p > 2^{32}$ are presented. One of these, $p = 188748146801$ for $a = 5$, also satisfies the ``reverse'' congruence $p^{a-1} \equiv 1 \pmod {a^2}$. An effective procedure for searching for such ``double solutions'' is described and applied to the range $a < 10^6$, $p <\max\, (10^{11}, a^2)$. Previous to this, congruences $a^{p-1} \equiv 1 \pmod {p^r}$ are generally considered for any $r \ge 2$ and fixed prime $p$ to see where the smallest prime solution $a$ occurs.

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Additional Information

Wilfrid Keller
Affiliation: Universität Hamburg, 20146 Hamburg, Germany

Jörg Richstein
Affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada

Keywords: Fermat quotient, Diophantine equation, primitive roots, large primes
Received by editor(s): July 30, 2001
Received by editor(s) in revised form: September 1, 2003
Published electronically: June 8, 2004
Additional Notes: The second author was supported by the Killam Trusts.
Article copyright: © Copyright 2004 American Mathematical Society