a) Let p be an odd prime and let
a = $\displaystyle \prod$ (2j-1) = (1)(3)(5)....(p-2)
Prove that $\displaystyle a^2$ = $\displaystyle (-1)^{(p+1)/2}$ mod p
(Product is from j=1 to (p-1)/2)
b) Let p be a prime. Prove that
$\displaystyle \left(\begin{array}{cc}2p\\p\end{array}\right)$ = 2 mod p
$\displaystyle \left(\begin{array}{cc}2p\\p\end{array}\right)$ is 2p choose p, as in statistics. so $\displaystyle (2p)!/p!(2p-p)!$