If i understand correctly, it sounds like you are asking if there are two different prime numbers p and q such that for integers m and n, p\^n =q\^m . The answer to this is no and this is basically because of how prime numbers are defined.

We say that an integer p is prime if for any a and b such p divides a\*b, then p divides at least one of a and b. For example 6 isn't prime because 2\*3=6, so it is divisible by 6, but 6 doesn't divide either 2 or 3.

So suppose there was an equation p\^n=q\^m, then we can write q\^m=q\*q\^m-1 and since q\^m =p\^n , it is divisible by p. However this means p must divide either q or q\^m-1, if p divides q then q is either equal to p or not prime. On the other hand if p divides q\^m-1, we can do the same thing, p must divide either q or q\^m-2 and so forth. So we conclude that p must divide q and then once again either q=p or q is not prime, so we are done.