# Integer solution to power equation

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• Dec 10th 2010, 05:35 PM
linshi
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?
• Dec 11th 2010, 10:33 AM
chiph588@
Quote:

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 \$.
• Dec 11th 2010, 10:40 AM
chiph588@
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!