\[p \gt 5\]

is a prime number. \[gcd(p,12)\]

must be equal to 1, so \[p \equiv 1, \; 5, \; 7, \; 11 \; (mod \; 12)\]

. If \[q \gt p \gt 5\]

is also prime then \[q \equiv 1, \; 5, \; 7, \; 11 \; (mod \; 12)\]

too.\[1^2=1, \; 5^2=25, \; 7^2=49, \; 11^2=121\]

The difference between any two of these is a multiple of 24, so

\[q^2-p^2 \equiv 0 \; (mod \; 24)\]

.