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Proceedings of the American Mathematical Society

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

 

 

Cauchy-type congruences for binomial coefficients


Authors: Richard H. Hudson and Kenneth S. Williams
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
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Abstract: In 1840 Cauchy [2] showed that for a prime $ p = ef + 1$, $ e = 20$,

$\displaystyle \left( {\begin{array}{* {20}{c}} {10f} \\ f \\ \end{array} } \rig... ...\begin{array}{* {20}{c}} {10f} \\ {3f} \\ \end{array} } \right)\quad (\bmod p),$

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$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0652435-9
Keywords: Davenport-Hasse relation, sign ambiguities in Cauchy-type congruences, binomial coefficients $ (\bmod p)$.
Article copyright: © Copyright 1982 American Mathematical Society