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This is being asked in the context of finding pairs of numbers that make the expression $a^b$ invariant to swapping exponent and base.

Let's say I have two distinct natural numbers $X,Y$, where:

$$\frac{X}{Y} = \frac{\ln(X)}{\ln(Y)}$$

Can this happen for only finitely many distinct $X,Y \in \mathbb{N}$

Here's a specific example I found:

$$(2,4): \ln(2)/\ln(4)=1/2$$

1 Answers1

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Write your equation as $\frac {\ln X}X=\frac {\ln Y}Y$ Logs are tiny compared to the number once the number gets at all large, so you need one of $X,Y$ to be less than $e$. There aren't many naturals to choose from. You can formalize this with derivatives of $\frac {\ln x}x$

Ross Millikan
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  • Thanks for the good intuition. I also found a nice paper on it from MAA some time ago: http://www.maa.org/sites/default/files/Sved50816668.pdf. –  Aug 01 '16 at 05:34
  • So, I'm concluding that $x^{y^z} = y^{z^x} = z^{x^y}$ doesn't have non-trivial integer solutions –  Aug 01 '16 at 05:41