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 .
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.