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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Solutions of the congruence $a^{p-1} \equiv 1 \pmod {p^r}$
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by Wilfrid Keller and Jörg Richstein PDF
Math. Comp. 74 (2005), 927-936 Request permission


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
  • Email:
  • Jörg Richstein
  • Affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada
  • Email:
  • 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.
  • © Copyright 2004 American Mathematical Society
  • Journal: Math. Comp. 74 (2005), 927-936
  • MSC (2000): Primary 11A07; Secondary 11D61, 11--04
  • DOI:
  • MathSciNet review: 2114655