Equation xʸ = yˣ

The equation ${\displaystyle x^{y}=y^{x}}$ is mentioned in a letter of Bernoulli to Goldbach (29 June 1728[2]). The letter contains a statement that when ${\displaystyle x\neq y,}$ the only solutions in natural numbers are ${\displaystyle (2,4)}$ and ${\displaystyle (4,2),}$ although there are infinitely many solutions in rational numbers, such as ${\displaystyle ({\tfrac {27}{8}},{\tfrac {9}{4}})}$ and ${\displaystyle ({\tfrac {9}{4}},{\tfrac {27}{8}})}$.[3][4] The reply by Goldbach (31 January 1729[2]) contains general solution of the equation, obtained by substituting ${\displaystyle y=vx.}$[3] A similar solution was found by Euler.[4]

J. van Hengel pointed out that if ${\displaystyle r,n}$ are positive integers with ${\displaystyle r\geq 3}$, then ${\displaystyle r^{r+n}>(r+n)^{r};}$ therefore it is enough to consider possibilities ${\displaystyle x=1}$ and ${\displaystyle x=2}$ in order to find solutions in natural numbers.[4][5]

The problem was discussed in a number of publications.[2][3][4] In 1960, the equation was among the questions on the William Lowell Putnam Competition,[6][7] which prompted Alvin Hausner to extend results to algebraic number fields.[3][8]

Main source:[1]

An infinite set of trivial solutions in positive real numbers is given by ${\displaystyle x=y.}$ Nontrivial solutions can be written explicitly as

${\displaystyle y=e^{-W_{-1}{\bigl (}{\frac {-\ln(x)}{x}}{\bigr )}}\quad \mathrm {for} \quad 1<x<e,}$

${\displaystyle y=e^{-W_{0}{\bigl (}{\frac {-\ln(x)}{x}}{\bigr )}}\quad \mathrm {for} \quad e<x.}$

Here, ${\displaystyle W_{-1}}$ and ${\displaystyle W_{0}}$ represent the negative and principal branches of the Lambert W function.

Nontrivial solutions can be more easily found by assuming ${\displaystyle x\neq y}$ and letting ${\displaystyle y=vx.}$ Then

${\displaystyle (vx)^{x}=x^{vx}=(x^{v})^{x}.}$

Raising both sides to the power ${\displaystyle {\tfrac {1}{x}}}$ and dividing by ${\displaystyle x}$, we get

${\displaystyle v=x^{v-1}.}$

Then nontrivial solutions in positive real numbers are expressed as

${\displaystyle x=v^{1/(v-1)},}$

${\displaystyle y=v^{v/(v-1)}.}$

Setting ${\displaystyle v=2}$ or ${\displaystyle v={\tfrac {1}{2}}}$ generates the nontrivial solution in positive integers, ${\displaystyle 4^{2}=2^{4}.}$

Other pairs consisting of algebraic numbers exist, such as ${\displaystyle {\sqrt {3}}}$ and ${\displaystyle 3{\sqrt {3}}}$, as well as ${\displaystyle {\sqrt[{3}]{4}}}$ and ${\displaystyle 4{\sqrt[{3}]{4}}}$.

The parameterization above leads to an interesting geometric property of this curve. It can be shown that ${\displaystyle x^{y}=y^{x}}$ describes the isocline curve where power functions of the form ${\displaystyle x^{v}}$ have slope ${\displaystyle v^{2}}$ for some positive real choice of ${\displaystyle v\neq 1}$. For example, ${\displaystyle x^{8}=y}$ has a slope of ${\displaystyle 8^{2}}$ at ${\displaystyle ({\sqrt[{7}]{8}},{\sqrt[{7}]{8}}^{8}),}$ which is also a point on the curve ${\displaystyle x^{y}=y^{x}.}$

The trivial and non-trivial solutions intersect when ${\displaystyle v=1}$. The equations above cannot be evaluated directly at ${\displaystyle v=1}$, but we can take the limit as ${\displaystyle v\to 1}$. This is most conveniently done by substituting ${\displaystyle v=1+1/n}$ and letting ${\displaystyle n\to \infty }$, so

${\displaystyle x=\lim _{v\to 1}v^{1/(v-1)}=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}=e.}$

Thus, the line ${\displaystyle y=x}$ and the curve for ${\displaystyle x^{y}-y^{x}=0,\,\,y\neq x}$ intersect at x = y = e.

As ${\displaystyle x\to \infty }$, the nontrivial solution asymptotes to the line ${\displaystyle y=1}$. A more complete asymptotic form is

${\displaystyle y=1+{\frac {\ln x}{x}}+{\frac {3}{2}}{\frac {(\ln x)^{2}}{x^{2}}}+\cdots .}$

• "Rational Solutions to x^y = y^x". CTK Wiki Math.
• "x^y = y^x - commuting powers". Arithmetical and Analytical Puzzles. Torsten Sillke. Archived from the original on 2015-12-28.
• dborkovitz (2012-01-29). "Parametric Graph of x^y=y^x". GeoGebra.
• OEIS sequence A073084 (Decimal expansion of -x, where x is the negative solution to the equation 2^x = x^2)