Hi!

We first rewrite the equality equivalently as .

If , we get , which forces and , so we've found one triplet.

So suppose from now on that is an odd prime.

If we have . We can't have because it implies . Forced is and , which gives us another triplet.

So suppose from now on that is an odd prime.

From we see that is an odd prime.

We now know , so we can rewrite as .

We cannot have (else wouldn't be a prime) so we must have and .

In other words, and .

Let for some . Then and .

If , we have , and , so we've found another triplet.

If , then which implies . So we cannot have .

If , then which implies , which we excluded, so we cannot have .

We conclude that the only triplets are 7,3,2 and 3,2,7 and 5,3,5.