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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Cauchy-type congruences for binomial coefficients
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by Richard H. Hudson and Kenneth S. Williams PDF
Proc. Amer. Math. Soc. 85 (1982), 169-174 Request permission

Abstract:

In 1840 Cauchy [2] showed that for a prime $p = ef + 1$, $e = 20$, \[ \left ( {\begin {array}{* {20}{c}} {10f} \\ f \\ \end {array} } \right ) \equiv \pm \left ( {\begin {array}{* {20}{c}} {10f} \\ {3f} \\ \end {array} } \right )\quad (\bmod p),\] and it was not until 1965 that Whiteman [6] succeeded in removing the sign ambiguity in this congruence. In this paper we show how the Davenport-Hasse relation [3] in the form given by Yamamoto [8] can be used to resolve the sign ambiguity in other Cauchy-type congruences. Details are given just for $e = 8,12,{\text {and }}20$.
References
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 85 (1982), 169-174
  • MSC: Primary 10A40; Secondary 05A10
  • DOI: https://doi.org/10.1090/S0002-9939-1982-0652435-9
  • MathSciNet review: 652435