# Thread: Integer solution to power equation

1. ## Integer solution to power equation

are the any integer solution to the equation
x^y = x y^x
other than x=0, y=0 or x=y=1?

I try to solve this by doing a substitution y=x^n (n is rational),
so it transforms to a polynomial-like equation
x^n - nx - 1 = 0.

but this seems not doing any help.

Any idea?

2. Originally Posted by linshi
are the any integer solution to the equation
x^y = x y^x
other than x=0, y=0 or x=y=1?

I try to solve this by doing a substitution y=x^n (n is rational),
so it transforms to a polynomial-like equation
x^n - nx - 1 = 0.

but this seems not doing any help.

Any idea?
Another obvious solution is $\displaystyle x = n, \; y = 1$.

3. I surmise there are no other solutions: assume $\displaystyle x,y > 1$

Informally, if $\displaystyle x < y$ then $\displaystyle x^{y-1} > y^x$, and if $\displaystyle y > x$ then $\displaystyle x^{y-1} < y^x$.

I say informally because this isn't always the case, but I believe there are only a finite number of cases where this fails.

*Someone correct me if I'm wrong here, I just glanced at this problem!