Generalisation of a congruence of Gauss
type: Item
instance of: scholarly article
title: Generalisation of a congruence of Gauss
zbMATH Open document ID: 0651.10004
DOI: 10.1016/0022-314X(88)90108-4
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: For a prime \(p\equiv 1(4)\), \(p=a^2+b^2\), \(a\equiv 1(4)\) unique, Gauss proved that \(\binom{(p-1)/2}{(p-1)/4}\equiv 2a\pmod p\). An improvement of this congruence is \(\binom{(p-1)/2}{(p-1)/4}\equiv (-4)^{(p-1)/4}(a+bi)\pmod{p^2}\), where \(a+bi\in\mathbb Z_p\) and \(| a+bi|_p=1\). This was proved by \textit{S. Chowla}, \textit{B. Dwork} and \textit{R. Evans} [J. Number Theory 24, 188--196 (1986; Zbl 0596.10003)]. Here the author generalizes this congruence to \[ \binom{(mp^{r-1})/2}{(mp^{r-1})/4}\equiv \binom{(mp^{r-1}-1)/2}{(mp^{r-1}-1)/4}(-4)^{\{mp^{r-1}\cdot (p-1)/4\}}\cdot (a+bi)\pmod {p^{2r}} \] where \(m\equiv 1(4)\). This congruence and many others are proved using Gauss and Jacobi sums and the theory of the \(p\)-adic \(\Gamma\)-function. The author remarks that these congruences are very easy to prove modulo \(p^r\) by the use of the theory of elliptic curves or the theory of formal group laws.
reviewed by: A. R. Rajwade
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11S80
zbMATH DE Number: 4061313
zbMATH Keywords: congruence of Gauss
zbMATH Keywords: binomial coefficients
zbMATH Keywords: Gauss sums
zbMATH Keywords: Jacobi sums
zbMATH Keywords: p-adic \(\Gamma\)-function
author: Matthijs J. Coster
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4061313