The rollback of my ignorance, however slight, is sufficient to complete my morning...

The rollback of my ignorance, however slight, is sufficient to complete my morning...  Friend and former colleague Doug W. (who actually understands the math that I dabble at the edges of) explains the “magic” p^2 - 1 factoid.  He writes:

You had a blog entry yesterday about the fact that p**2 - 1 is divisible by 24 for any prime p > 3. It's actually very simple to derive the result. Assume that p is an odd number, not divisible by 3 (in the problem as stated, we know that p isn't divisible by 3, since p is prime :-).

Now, p**2 - 1 = (p+1)(p-1). Note that these two factors are consecutive even integers. Thus, exactly one of them must be multiple of 4, so when you multiply the two factors together you get a multiple of 8. That is, we know that p-1 is either of the form 4k or 4k+2, in which case p+1 would be of the form 4k+2 or 4k+4, respectively. So the product is either of the form 4k*(4k+2) or (4k+2)(4k+4), each of which is divisible by 8.

Further, since p is not divisible by 3, and since in any three consecutive integers there is exactly one multiple of 3, then either (p+1) or (p-1) must be divisible by 3. Thus, the overall product is a multiple of 8*3 (i.e., 24).

He's right, it is simple!  Assuming, that is, you're facile enough with math to be able to work it out :)