Note,

76=2*38

34=2*17

And m<n otherwise we have a negative number.

2^m*38^m-2^n*17^n=1039845338

Thus, the left hand side is divisible by 2^m

But the right hand side is divisible at most by 2 (not by 4=2^2).

Thus, this forces m=1.

In that case you can attempt to solve this equation for n since you determined what m has to be.