Can there exist three integers p, q and r with 2 ≤ p < q < r, that satisfy each of the following conditions?
(i) p2 -1 is divisible by each of q and r, and:
(ii) r2 -1 is divisible by each of p and q.
No, this is not possible.
Lemma) x divides y2 -1 only if x and y are relatively prime. This is because if they had any common factors, that common factor would divide y2 evenly, so it could not divide y2 -1 evenly.
Proof) Assume that p, q and r exists.
Since r2 -1 is divisible by q, q and r do not share a common factor.
Since q & r both divide p2 -1, and since they have no common factors, then q*r <= p2 -1, since they at most have all factors of p2 -1 accounted for between them, without any overlap. But q and r are both > p, so q*r >p*p > p2 -1, so a contradiction exists.
Therefore, p, q and r do not exist.
Relatively speaking (solution)
