Show that an integer $p > 3$ is a prime if and only if for every two nonzero integers $a,b$ exactly one of the numbers $N_1 = a+b-6ab+\frac{p-1}{6}$ , $N_2 = a+b+6ab+\frac{p-1}{6}$ is a nonzero integer.
Source: Czech and Slovak Match 1996 P1
Tags: prime, Integers, number theory
Show that an integer $p > 3$ is a prime if and only if for every two nonzero integers $a,b$ exactly one of the numbers $N_1 = a+b-6ab+\frac{p-1}{6}$ , $N_2 = a+b+6ab+\frac{p-1}{6}$ is a nonzero integer.