Rational Number as Power of Number with Itself

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Theorem

Every rational number in the interval $\left({\left({\dfrac 1 e}\right)^{\frac 1 e} \,.\,.\, +\infty}\right)$ can be written either as:

$a^a$ for some irrational number $a$

or as:

$n^n$ for some natural number $n$.


Proof


Sources