For math professionals. Prove that this Diophantine equation has only one answers.?

Question

Prove that the answer of this equation is only x=y=z=2

3^x+4^y=5^z

Answer 1

Step 1) Mod3:

0 + 1^y = 2^z = 1 Mod 3. Therefore z is even. Set z=2u.

Step 2) Mod4:

(-1)^x + 0 = (1)^z Mod4. Therefore x is even. Set x=2v.

Step 3) 5^2u - 3^2v = 4^y = (5^u + 3^v)(5^u - 3^v)

Here, the difference of the two factors on the right is 2*3^u

which means not both factors are divisible by 4, yet each is a

power of 2 forcing, 5^u - 3^v = 2.

Step 4) (5^u + 3^v)/(5^u - 3^v) = 1 + 2*3^v/(5^u - 3^v) =

1 + 3^v = 2^r since both 5^u + 3^v and 5^u - 3^v are

powers of 2.

Step 4) 1 + 3^v = 2^r and 2^r = 1 Mod3 gives even r = 2s.

Therefore, 3^v = 2^(2s) - 1 = (2^s + 1)(2^s - 1) but these

factors differ by 2 and therefore not both are divisible by 3.

This forces 2^s - 1 = 1 ans s=1, then r = 2, v = 1, and then

u = 1 in quick succession.

Step 5) Recalling x = 2v and z = 2u gives x = z = 2 and

finally y = 2 as required.

Answer 2

Just a comment on Kyle's answer.
It's not quite accurate:
x,y, z can't be all equal and greater than 2.
That's Fermat's Last Theorem. It is
possible they could all be different and greater than 2.
Also, Pretzels, this is a Diophantine equation,
so we are looking for integer solutions.
I'll also assume that x,y,z are positive integers.
A good place to start might be to write
5^z-4^y = 3^x
and divide by 4^y to get
5^z/4^y - 1 = 3^x/4^y.
So there is no solution if
5^z/4^y >=2
and
3^x/4^y <1.
Also, since z >= 2,
25 must divide 3^x+4^y.
As I said, this is just a start and I'll play with these
ideas and see what happens.

Answer 3

Well i know how to prove that they aren't all greater than 2. That would be Fermat's Last Theorem. There rest...no idea. Great question. Hopefully someone can give the complete proof.


Pretzels:
With your solution, the equation would be this:
7 = 6.99999999...99599...
Doesn't quite work because of that 5.

Answer 4

This equation has any number of solutions.

For example: x=1, y=1, z=ln(7)/ln(5) ~ 1.209

If you restrict x, y , and z to the integers then it may have only one. solution