Difference of Squares of Odd Primes Modulus 24
\[p \gt 5\]
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
\[q^2-p^2 \equiv 0 \; (mod \; 24)\]
. 
