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Find all rational solutions to $m^n = n^m$, where $|m| > |n|$.

I know how to solve the problem when $m$ and $n$ are integers, and the easiest method in my opinion is to rewrite the equation as $\frac{1}m \log m = \frac{1}n \log n (1)$. I think it's useful to first consider the case where m and n are positive rational numbers and $m < n$. Then for equality to hold for (1), we need $m < e, n > e$ (because $\frac{1}x\log x$ is strictly increasing for $0 < x \leq e$ and strictly increasing for $x > e$). Clearly the only integer solutions satisfying the requirement are $m = 2, n = 4$. But what about rational solutions?

Gord452
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  • Note: that duplicate was (in my view, inappropriately) closed as a duplicate of a question asking for integer solutions. But one of the posted solutions at least contains an infinite family of rational solutions. – lulu Jun 23 '22 at 13:57
  • Let $n=\frac{a}{b}$, $m=\frac{c}{d}$, $a,b,c,d\in\mathbb{N}$, $gcd(a,b)=gcd(c,d)=1$. Then $m^n=n^m$ can be rewritten as $a^{bc}d^{ad}=b^{bc}c^{ad}$. As $gcd(a,b)=gcd(c,d)=1$, numbers $a$ and $c$ have the same prime factors, and numbers $b$ and $d$ have the same prime factors. Then $a^{bc}=c^{ad}$, $b^{bc}=d^{ad}$. – Ivan Kaznacheyeu Jun 23 '22 at 14:03
  • Let $gcd(ad,bc)=gcd(a,c) gcd(b,d)=k$, $ad=kl$, $bc=km$ (another $m$), $gcd(l,m)=1$. Then $a^m=c^l$, $a=r^l$, $c=r^m$, $b^m=d^l$, $b=s^l$, $d=s^m$. Let WLOG $l\geq m$, then $k=gcd(ad,bc)=(rs)^m$, $ad=kl \Rightarrow r^l s^m = l (rs)^m \Rightarrow r^{l-m}=l$, $bc=km\Rightarrow r^m s^l=m (rs)^m \Rightarrow s^{l-m}=m$. – Ivan Kaznacheyeu Jun 23 '22 at 14:10
  • $l>m$, $r^{l-m}=l$, $s^{l-m}=m$ $\Rightarrow r>s \Rightarrow r\geq s+1$. $l-m=r^{l-m}-s^{l-m}\geq (s+1)^{l-m}-s^{l-m}\geq (l-m)s^{l-m-1}$ $\Rightarrow s^{l-m-1}\leq 1 \Rightarrow l-m=1 \lor s=1$. $s=1 \Rightarrow b=d=1$, integer case. $l-m=1 \Rightarrow$ $r=m+1,s=m$, $a=(m+1)^{m+1}$, $b=m^{m+1}$, $c=(m+1)^m$, $d=m^m$. – Ivan Kaznacheyeu Jun 23 '22 at 14:26
  • Example $(\frac{27}{8})^{\frac{9}{4}}=(\frac{9}{4})^{\frac{27}{8}}$ – Ivan Kaznacheyeu Jun 23 '22 at 14:27

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