Let a=(p+1)/2, b=(p-1)/2.

Unless p is 2, p is the difference between a² and b²:

a² - b² = (p²+2p+1)/4 - (p²-2p+1)/4 = 4p/4 = p.

Thanks to Guy de Kindler for this one, to Nick Hobson for showing that this property to true not only of primes but odd numbers and numbers evenly divisible by four, and to Terry Ryder for pointing out some previous errors.