AN ELEMENTARY APPROACH TO $\binom{(p-1)/2}{(p-1)/4}$ modulo $p^2$

Abstract

We give an elementary proof of the well-known congruence $$ \binom{\frac{p-1}{2}}{\frac{p-1}{4}} \equiv \frac{2^{p-1}+1}{2} \bigg(2a-\frac{p}{2a}\bigg) \pmod{p^2}, $$ where $p \equiv 1 \pmod{4}$ is prime and $p = a^2 + b^2$ with $a \equiv 1 \pmod{4}$.