a) Letpbe 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) Letpbe 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 $\displaystyle (2p)!/p!(2p-p)!$