Prime Square Divisibility
Prove that p21 is divisible by 24 for all primes, p > 3.
For all primes greater than 3, p = 6k1.
When p = 6k+1, p2 = 36k2 + 12k + 1 and so p21 = 36k2 + 12k.
Similarly, when p = 6k1, p21 = 36k2 12k.
Therefore, p21 = 36k2 12k = 12k(3k 1).
If k is odd, 3k 1 will be even and so we prove that p21 will always be divisible by 12 2 = 24.
What can you say about p31?
What about other powers of p?