Hi,

I wonder whether somebody can advise me how to solve in natural numbers (or integers) the following equation:

$\displaystyle 3^x+4^y=5^z$

I have spent a couple of weeks thinking about it without any considerable progress

. The obvious guess is that the only solution is (x,y,z)=(2,2,2). I've looked at the equation modulo different numbers 3,4,5,9,16,25 ... etc, but so far that wasn't very conclusive in proving that (2,2,2) is the only solution. I've tried a lot of other different things with the equation with the same negative result. Honestly I've run out of ideas of how to solve the equation.

Any help is appreciated.