Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
MathSciNet review: 652435
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • [1] Bruce C. Berndt and Ronald J. Evans, Sums of Gauss, Jacobi, and Jacobsthal, J. Number Theory 11 (1979), no. 3, S. Chowla Anniversary Issue, 349–398. MR 544263, 10.1016/0022-314X(79)90008-8
  • [2] A. Cauchy, Mémoire sur la théorie des nombres, Mém. Institut de France 17 (1840), 249-278 (Ouevres completés (1) 3 (1911), 5-83).
  • [3] H. Davenport and H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. Reine Angew. Math. 172 (1934), 151-182.
  • [4] C. F. Gauss, Theoria residuorum biquadraticorum, Werke, vol. 2, p. 90.
  • [5] Emma Lehmer, Criteria for cubic and quartic residuacity, Mathematika 5 (1958), 20–29. MR 0095162
  • [6] Albert Leon Whiteman, Theorems on Brewer and Jacobsthal sums. I, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 44–55. MR 0175861
  • [7] Albert Leon Whiteman, Theorems on Brewer and Jacobsthal sums. II, Michigan Math. J. 12 (1965), 65–80. MR 0217037
  • [8] Koichi Yamamoto, On a conjecture of Hasse concerning multiplicative relations of Gaussian sums, J. Combinatorial Theory 1 (1966), 476–489. MR 0213311

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 10A40, 05A10

Retrieve articles in all journals with MSC: 10A40, 05A10


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