x and y must be greater than 2, and both odd, so their difference must be two.

If z = 2, then y-2 and y and x are all prime.

The only prime triplet is 3,5,7, so x = 7 and y = 5

What if z is not 2?

Then x, y and z are all odd.

Then x - y = 2 and y - z = 2, so x - z = 4, which is not prime.

So there is only 1 solution.