Prime numbers are of the form p = 6k + 1 or p = 6k - 1.

If p = 6k + 1, 8p + 1 = 48k + 9 = 3(16k + 3). Thus 8p + 1 has 3 as factor. 8p + 1 = 3 is a contradiction since p is prime.

Couldn't figure out p = 6k - 1 yet, but maybe you could do it this way?

EDIT EDIT EDIT:

Ok, I think I figured it out.

Consider p = 6k - 1. Now 8p - 1 = 6l + 1 or 8p - 1 = 6l - 1.

If 8p - 1 = 6l + 1, 8p + 1 = 6l + 3 = 3(2l + 1) and 8p + 1 has 3 as a factor and thus is composite.

If 8p - 1 = 6l - 1, 8p = 6l and p = 3l / 4 = 3 * (l / 4). Now this means p has three as a factor and thus p = 3. Therefore 8p + 1 = 8*3 + 1 = 25 which is composite.

Does anyone see anything wrong with this proof?

EDIT 2: Oops, fixed a mistake. 8p + 1 = 48k + 9, not 8p + 1 = 48k + 8.