An elementary approach to \(\binom{(p-1)/2}{(p-1)/4}\) modulo \(p^2\)
type: Item
instance of: scholarly article
title: An elementary approach to \(\binom{(p-1)/2}{(p-1)/4}\) modulo \(p^2\)
zbMATH Open document ID: 1275.11005
author: Hao Pan
published in: Taiwanese Journal of Mathematics
publication date: 2013-02-21T00:00:00Z
full work available at URL: http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/view/2637
review text: This note gives an elementary proof of the following congruence \[ \binom{(p-1)/2}{(p-1)/4}\equiv \frac{2^{p-1}+1}{2}\left(2a-\frac{p}{2a}\right)\pmod {p^ 2}, \] where \(p\equiv 1\pmod 4\) is prime and \(p=a^ 2+b^ 2\) with \(a\equiv 1\pmod 4\). The author recalls the origin of such result as a generalisation of the well-known congruence of Gauss \( \binom{(p-1)/2}{(p-1)/4}\equiv 2a \pmod p\) conjectured by \textit{F. Beukers} and established by \textit{S. Chowla, B. Dwork} and \textit{R. Evans} [J. Number Theory 24, 188--196 (1986; Zbl 0596.10003)] who connected the Gauss sums to the \(p\)-adic \(\Gamma\)-functions via the Gross-Koblitz formula. The author remarks the simplicity of his alternative approach which employs only the Chu-Vandermonde identity and a basic formula from [\textit{B. C. Berndt, R. J. Evans} and \textit{K. S. Williams}, Gauss and Jacobi sums. New York, NY: John Wiley (1998; Zbl 0906.11001)].
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11B65
zbMATH DE Number: 6137998
zbMATH Keywords: binomial coefficients
zbMATH Keywords: Chu-Vandermonde identity
zbMATH Keywords: Legendre symbol
MaRDI profile type: Publication
OpenAlex ID: W4254074485
DOI: 10.11650/TWJM/1500406847
description: scientific article; zbMATH DE number 6137998