Integer solutions for $x^y=y^x$

Asked Mar 07 '18 at 14:41

Active Mar 07 '18 at 15:08

Viewed 66 times

Let's say we have the equation: $$x^y=y^x$$ what are the solutions when $x,y \in \Bbb Z$ for ${x}\ne{y}$..

The only one I have found is 2 and 4. There are infinite non-integer solutions but I am interested in the integer ones. Also you can find a new definition for $e$ from this equation $e=x^{\frac{y}{xln(y)}}$ and also $log_x(y)=\frac{y}{x}$

edited Mar 07 '18 at 15:08

asked Mar 07 '18 at 14:41

Latin Wolf

1

There is $1$... – Andrew Li Mar 07 '18 at 14:43
3

Presumably you require $x\ne y$? – almagest Mar 07 '18 at 14:45
See also this question, and others linked to it. – Dietrich Burde Mar 07 '18 at 14:49

Integer solutions for $x^y=y^x$

0 Answers0