There is only one solution: .

Here's a somewhat messy proof (I'm sure that a number theorist could come up with a better one).

I'll write 2n in place of n, just to keep track of the fact that n is even. So p and q are primes, and .

Therefore ,

,

.

Thus q divides , say , where and . Then and so .

But kp > p–1 and therefore , so that . This is only possible if k=1. Therefore .

If then , so and therefore . If p is odd then this means that q is even, which is not possible since q is clearly greater than p. Therefore p=2 and q=5.

If then a similar calculation to the previous paragraph gives which is obviously impossible.

So the only solution is p=2, q=5.