Squared Primes (almost)

All primes $\gt 3$ are of the form $6n \pm 1$ for some positive integer $n$ , in which case $p^{2} - 1 = (6n \pm 1)^{2} - 1 = 36n^{2} \pm 12n = 12n(3n \pm 1)$ .

Now since $5^{2} - 1 = 24$ the maximum possible gcf is $24$ . In general, if $n$ is even then $24$ divides $12n$ , and if $n$ is odd then $3n \pm 1$ is even and so $24$ divides $12(3n \pm 1)$ . So whether $n$ is even or odd, $24$ divides $p^{2} - 1$ for any prime $p \gt 3$ , and since $24$ is the maximum possible gcf, the desired answer is $\boxed{24}$ .