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 $a \equiv 1 \pmod{4}$.
References
B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi sums, Wiley, New York, 1998. 0906.11001 B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi sums, Wiley, New York, 1998. 0906.11001
S. Chowla, B. Dwork and R. Evans, On the mod $p\sp 2$ determination of $\binom{(p-1)/2}{(p-1)/4}$, J. Number Theory, 24 (1986), 188-196. 0596.10003 10.1016/0022-314X(86)90102-2 S. Chowla, B. Dwork and R. Evans, On the mod $p\sp 2$ determination of $\binom{(p-1)/2}{(p-1)/4}$, J. Number Theory, 24 (1986), 188-196. 0596.10003 10.1016/0022-314X(86)90102-2
