We have x^{y} = y^{x}, where x and y are positive real numbers, with x < y.

If we set y = rx, where r > 1, we can derive a parametric solution: x = r^{1/(r−1)}, y = r^{r/(r−1)}.

Equivalently, setting r = 1 + 1/k, where k > 0, we have x = (1 + 1/k)^{k}, y = (1 + 1/k)^{k+1}.

Real solutions exist only for x > 1.

The only rational solutions are those for which, in the above formula, k is a positive integer.

That is, all rational solutions are of the form x = (1 + 1/n)^{n}, y = (1 + 1/n)^{n+1}, where n = 1, 2, ... .