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Answer to puzzle 48: Exponential equation

We have xy = yx, 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 = r1/(r−1), y = rr/(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, ... .