Thread: Equation with exponents

Could you help me solve the following equation:



where are integers.

I know the only solutions are (3,0,3) and (2,1,3). The problem is - I don't know how to prove it. Please help.

I understand your problem well. You can find an expression for x,y and k.







Similarly,

Thanks, but could you tell me where do those three equations lead to?

y=0,1
k=3
Therefore x=ln(9-5)/ln(2)=2
And x=ln(9-1)/ln(2)=3

Similarly find the values for y and k. But remenber you cant find the values of three variables from just 1 equation.

Thank you very much for the hint. Could you please take a quick look at what I came up with as I was waiting for your reply?

If y=0, we have , so and are powers of 2, and the only powers of 2 that differ by 2 are 2 and 4, so (x,y,k)=(3,0,3).

If y>0, is a quadratic residue modulo 5, so x must be even. Hence . The factors are powers of 5, say and .

We now have . Since the right side is not divisible by 5, we must have n=0(is it correct?). Then, for m=1, we get x=2, and so (x,y,k)=(2,1,3). But what next? How do I prove that there are no more solutions?