8 divides   n^2 - 1 for odd n

Proof:

Let n = 2m+1. The n^2 -1 = 4m^2+4m = 4k. So m^2 + m = k. Now if m is even, k=2s. If m is odd say 2r+1, then 4r^2+4r+1 + 2r+1 = 2s. So k is an even number = 2t so that n^2 - 1 = 4*2t = 8t as stated.

Cino Hilliard