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While talking with a friend about the number $16$, I began thinking about what two positive rational numbers, $n$ and $m$, satisfy the condition that $n^m$ = $m^n$ where $n$ and $m$ are not equal. $2$ and $4$, as discussed with my friend, are the first pair of integers to do this, and seem to be the only pair to do this after brute forcing combinations in a python script. It seems like there should be more possible pairs, so are $2$ and $4$ the only positive rational numbers to satisfy these conditions?

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Consider taking the $mn$-th root $$n^m=m^n\\n^{1/n}=m^{1/m}$$ Now the function $f(x)=x^{1/x}$ attains it's maximum at $x=e$ and is increasing from $(0,e)$ and decreasing from $(e,\infty)$ WLOG assume $m>n$ then $m\in (e,\infty)$ while $n\in (0,e)$.

If we limit $m,n$ to be integers we can see that either $m=1$ or $m=2$ which both give the trivial solutions $m=n=1$ and $m=2,n=4$.

In general if we are looking for rational solutions we get the parametric equation by setting $n=km$ $$m^{km}=(km)^m\\m^{(k-1)m}=k^m\\m=k^{1/(k-1)}$$ From this putting $t=\frac{1}{k-1}$ we get $m=(1+\frac1t)^t$ and $n=(1+\frac{1}{t})^{t+1}$,this parametrization gives infinitely many rational solutions (for any integer $t\neq -1,0$) for example for $t=2$ we get $2.25^{3.375}=3.375^{2.25}$

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We can state the requirement $m^n=n^m$ as $$ \frac{\ln n}{n}=\frac{\ln m}{m} $$ We can define $f(x)=\frac{\ln x}{x}$ for $x>0$. This function is increasing for $x<e$ and decreasing for $x>e$. It attains negative values for $x<1$ and positive values for $x>1$. Therefore, the only possible way of finding two values $x_1$ and $x_2$ such that $x_1<x_2$ and $f(x_1)=f(x_2)$ is to choose $x_1\in (1,e)$ and $x_2\in(e,\infty)$. Therefore $x_1 = 2$ and $x_2=4$ is the only solution with integers.

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    $\begingroup$ The OP is asking about rational solutions, not just integer ones. $\endgroup$ May 3, 2017 at 21:02
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Take log's of both sides and un-cross multiply and your equation becomes

$$\frac{\ln m}{m} = \frac{\ln n}{n}.$$

The graph of the function $f(x) = \ln(x)/x$ tells the story. $f$ has a local max at $x=e$. Any horizontal line $y = k$ with $0<k<1/e$ intersects the graph at two points. The $x$-coordinates of those two points are a solution to your equation.

Take $k=1/5$. Maple gives the two solutions in terms of Lambert Omega functions, and gives decimal approximations: $1.295855509, 12.71320679.$ So there are lots of real solutions, but none as pretty as $2, 4$.

Pick a rational number $a/b$ between $1$ and $e$. Set $\ln(x)/x = \ln(a/b)/(a/b)$. One solution is rational $x=a/b$. The other solution is given by $$ \Omega\left(-1,\frac{b}{a}\ln \frac{b}{a}\right)\frac{\frac{a}{b}}{\ln\frac{b}{a}}.$$

This seems unlikely to be rational. Maybe look at the series expansion of $\Omega(\ln(x))$(?)

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  • $\begingroup$ Are the exact values both rational? $\endgroup$ May 3, 2017 at 21:03
  • $\begingroup$ I don't think so. The first one is $-5\Omega(-1/5).$ The other one is just the point on the other branch. It's an interesting question: When is the Lambert function rational for rational inputs? $\endgroup$
    – B. Goddard
    May 3, 2017 at 21:06
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    $\begingroup$ The question asks for rational pairs. $\endgroup$ May 3, 2017 at 21:07
  • $\begingroup$ @DanielFischer Questions get edited. $\endgroup$
    – B. Goddard
    May 3, 2017 at 21:19
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    $\begingroup$ They do. But this one had "rational" in the body from the beginning. $\endgroup$ May 3, 2017 at 21:21

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