An elementary approach to \(\binom{(p-1)/2}{(p-1)/4}\) modulo \(p^2\)
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)].
scientific article; zbMATH DE number 6137998