If you have some more information on m and n, there might be something more to say, but for general m and n, the only thing you can guarantee is 2.
Is this part of a larger problem?
Ok, I see now.
So your conjecture is that gcd(5^m + 7^m,5^n + 7^n) is sometimes 2 and sometimes 5^gcd(m,n) + 7^gcd(m,n) but never any other value, and you don't yet have a choice function to tell which one it is. That's an improvement on what I had to say. I said essentially that it's 2 or some multiple of 2.
For what it's worth, I can confirm your conjecture for all m,n less than 1000.
I was waiting for the original poster to say something.
It depends on the factorization
which works if n/r is odd. So have a common factor if r is a common factor of n and m, and n/r and m/r are both odd.
So in the prime factorizations of n, m, and r, the power of 2 is the same. In fact, we should have:
if the prime factorizations of n and m contain the same power of 2.
I'm still foggy on the converse, though. Couldn't there be a common factor that's not of the form ?