Cauchy-type congruences for binomial coefficients
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- by Richard H. Hudson and Kenneth S. Williams
- Proc. Amer. Math. Soc. 85 (1982), 169-174
- DOI: https://doi.org/10.1090/S0002-9939-1982-0652435-9
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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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Bibliographic 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