Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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 $a^{p-1} \equiv 1 \pmod {p^r}$
HTML articles powered by AMS MathViewer

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.
Similar Articles
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