Primes

Note that if $p \equiv 1,2 \pmod{3}$ , $8p^2+1 \equiv 0 \pmod{3}$ . Hence, we must have, $p \equiv 0 \pmod{3}$ . And, since p is also a prime, we are left with only 1 possibility, i.e. $p=3$ . Now, we need to check whether this is a solution or not. So, when $p=3$ , $8p^2+1=73$ , which is a prime.