Thread: 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?

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 .

I surmise there are no other solutions: assume

Informally, if then , and if then .

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!