xy=yxx^y = y^x for integers xx and yy

We know that 24=42 and (2)4=(4)2. Is there another pair of integers x,y (xy) which satisfies the equality xy=yx?

Answer

This is a classic (and well known problem).

The general solution of xy=yx is given by

x=(1+1/u)uy=(1+1/u)u+1

It can be shown that if x and y are rational, then u must be an integer.

For more details, see this and this.

Attribution
Source : Link , Question Author : Paulo Argolo , Answer Author : J. M. ain’t a mathematician

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