The key point here is that 2 is a quadratic residue modulo an odd prime p if and only if p is congruent to 1 or 7 (mod 8). Therefore cannot have any prime divisors of the form 8k+3 or 8k+5. It clearly is divisible by 2, but all its other prime divisors must be congruent to 1 or 7 (mod 8), and at least one of them must be congruent to 7 (mod 8) (otherwise their product would be congruent to 2 (mod 16)).