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In general, exponentiation fails to be commutative. However, the equation ${\displaystyle x^{y}=y^{x}}$ has solutions, such as ${\displaystyle x=2,\ y=4.}$

History

The equation ${\displaystyle x^{y}=y^{x}}$ is mentioned in a letter of Bernoulli to Goldbach (29 June 1728). 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 ${\displaystyle ({\tfrac {9}{4}},{\tfrac {27}{8}})}$. The reply by Goldbach (31 January 1729) contains a general solution of the equation, obtained by substituting ${\displaystyle y=vx.}$ A similar solution was found by Euler.

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.

The problem was discussed in a number of publications. In 1960, the equation was among the questions on the William Lowell Putnam Competition, which prompted Alvin Hausner to extend results to algebraic number fields.

Positive real solutions

Main source:

Explicit form

An infinite set of trivial solutions in positive real numbers is given by ${\displaystyle x=y.}$ Nontrivial solutions can be written explicitly using the Lambert W function. The idea is to write the equation as ${\displaystyle ae^{b}=c}$ and try to match ${\displaystyle a}$ and ${\displaystyle b}$ by multiplying and raising both sides by the same value. Then apply the definition of the Lambert W function ${\displaystyle a'e^{a'}=c'\Rightarrow a'=W(c')}$ to isolate the desired variable.

${\displaystyle {\begin{aligned}y^{x}&=x^{y}=\exp \left(y\ln x\right)&\\y^{x}\exp \left(-y\ln x\right)&=1&\left({\mbox{multiply by }}\exp \left(-y\ln x\right)\right)\\y\exp \left(-y{\frac {\ln x}{x}}\right)&=1&\left({\mbox{raise by }}1/x\right)\\-y{\frac {\ln x}{x}}\exp \left(-y{\frac {\ln x}{x}}\right)&={\frac {-\ln x}{x}}&\left({\mbox{multiply by }}{\frac {-\ln x}{x}}\right)\end{aligned}}}$

${\displaystyle \Rightarrow -y{\frac {\ln x}{x}}=W\left({\frac {-\ln x}{x}}\right)}$

${\displaystyle \Rightarrow y={\frac {-x}{\ln x}}\cdot W\left({\frac {-\ln x}{x}}\right)=\exp \left(-W\left({\frac {-\ln x}{x}}\right)\right)}$

Where in the last step we used the identity ${\displaystyle W(x)/x=\exp(-W(x))}$.

Here we split the solution into the two branches of the Lambert W function and focus on each interval of interest, applying the identities:

${\displaystyle {\begin{aligned}W_{0}\left({\frac {-\ln x}{x}}\right)&=-\ln x\quad &{\text{for }}&0<x\leq e,\\W_{-1}\left({\frac {-\ln x}{x}}\right)&=-\ln x\quad &{\text{for }}&x\geq e.\end{aligned}}}$

• ${\displaystyle 0<x\leq 1}$:

${\displaystyle \Rightarrow {\frac {-\ln x}{x}}\geq 0}$

${\displaystyle {\begin{aligned}\Rightarrow y&=\exp \left(-W_{0}\left({\frac {-\ln x}{x}}\right)\right)\\&=\exp \left(-(-\ln x)\right)\\&=x\end{aligned}}}$

• ${\displaystyle 1<x<e}$:

${\displaystyle \Rightarrow {\frac {-1}{e}}<{\frac {-\ln x}{x}}<0}$

${\displaystyle \Rightarrow y={\begin{cases}\exp \left(-W_{0}\left({\frac {-\ln x}{x}}\right)\right)=x\\\exp \left(-W_{-1}\left({\frac {-\ln x}{x}}\right)\right)\end{cases}}}$

• ${\displaystyle x=e}$:

${\displaystyle \Rightarrow {\frac {-\ln x}{x}}={\frac {-1}{e}}}$

${\displaystyle \Rightarrow y={\begin{cases}\exp \left(-W_{0}\left({\frac {-\ln x}{x}}\right)\right)=x\\\exp \left(-W_{-1}\left({\frac {-\ln x}{x}}\right)\right)=x\end{cases}}}$

• ${\displaystyle x>e}$:

${\displaystyle \Rightarrow {\frac {-1}{e}}<{\frac {-\ln x}{x}}<0}$

${\displaystyle \Rightarrow y={\begin{cases}\exp \left(-W_{0}\left({\frac {-\ln x}{x}}\right)\right)\\\exp \left(-W_{-1}\left({\frac {-\ln x}{x}}\right)\right)=x\end{cases}}}$

Hence the non-trivial solutions are:

${\displaystyle y={\begin{cases}\exp \left(-W_{0}\left({\frac {-\ln(x)}{x}}\right)\right)\quad &{\text{for }}x>e,\\\exp \left(-W_{-1}\left({\frac {-\ln x}{x}}\right)\right)\quad &{\text{for }}1<x<e.\end{cases}}}$

Parametric form

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 the parametric equation

${\displaystyle {\begin{aligned}x&=v^{1/(v-1)},\\y&=v^{v/(v-1)}.\end{aligned}}}$

The full solution thus is ${\displaystyle (y=x)\cup \left(v^{1/(v-1)},v^{v/(v-1)}\right){\text{ for }}v>0,v\neq 1.}$

Based on the above solution, the derivative ${\displaystyle dy/dx}$ is ${\displaystyle 1}$ for the ${\displaystyle (x,y)}$ pairs on the line ${\displaystyle y=x,}$ and for the other ${\displaystyle (x,y)}$ pairs can be found by ${\displaystyle (dy/dv)/(dx/dv),}$ which straightforward calculus gives as:

${\displaystyle {\frac {dy}{dx}}=v^{2}\left({\frac {v-1-\ln v}{v-1-v\ln v}}\right)}$

for ${\displaystyle v>0}$ and ${\displaystyle v\neq 1.}$

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 ${\displaystyle {\sqrt{4}}}$ and ${\displaystyle 4{\sqrt{4}}}$.

The parameterization above leads to a 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 ${\displaystyle ({\sqrt{8}},{\sqrt{8}}^{8}),}$ 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

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

Other real solutions

An infinite set of discrete real solutions with at least one of ${\displaystyle x}$ and ${\displaystyle y}$ negative also exist. These are provided by the above parameterization when the values generated are real. For example, ${\displaystyle x={\frac {1}{\sqrt{-2}}}}$, ${\displaystyle y={\frac {-2}{\sqrt{-2}}}}$ is a solution (using the real cube root of ${\displaystyle -2}$). Similarly an infinite set of discrete solutions is given by the trivial solution ${\displaystyle y=x}$ for ${\displaystyle x<0}$ when ${\displaystyle x^{x}}$ is real; for example ${\displaystyle x=y=-1}$.

Similar graphs

Equation √y = √x

The equation ${\displaystyle {\sqrt{y}}={\sqrt{x}}}$ produces a graph where the line and curve intersect at ${\displaystyle 1/e}$. The curve also terminates at (0, 1) and (1, 0), instead of continuing on to infinity.

The curved section can be written explicitly as

${\displaystyle y=e^{W_{0}(\ln(x^{x}))}\quad \mathrm {for} \quad 0<x<1/e,}$

${\displaystyle y=e^{W_{-1}(\ln(x^{x}))}\quad \mathrm {for} \quad 1/e<x<1.}$

This equation describes the isocline curve where power functions have slope 1, analogous to the geometric property of ${\displaystyle x^{y}=y^{x}}$ described above.

The equation is equivalent to ${\displaystyle y^{y}=x^{x},}$ as can be seen by raising both sides to the power ${\displaystyle xy.}$ Equivalently, this can also be shown to demonstrate that the equation ${\displaystyle {\sqrt{y}}={\sqrt{x}}}$ is equivalent to ${\displaystyle x^{y}=y^{x}}$.

Equation log_x(y) = log_y(x)

The equation ${\displaystyle \log _{x}(y)=\log _{y}(x)}$ produces a graph where the curve and line intersect at (1, 1). The curve becomes asymptotic to 0, as opposed to 1; it is, in fact, the positive section of y = 1/x.

References

1. ^ Lóczi, Lajos. "On commutative and associative powers". KöMaL. Archived from the original on 2002-10-15. Translation of: "Mikor kommutatív, illetve asszociatív a hatványozás?" (in Hungarian). Archived from the original on 2016-05-06.
2. ^ Singmaster, David. "Sources in recreational mathematics: an annotated bibliography. 8th preliminary edition". Archived from the original on April 16, 2004.{{cite web}}: CS1 maint: unfit URL (link)
3. ^ Sved, Marta (1990). "On the Rational Solutions of x = y" (PDF). Mathematics Magazine. 63: 30–33. doi:10.1080/0025570X.1990.11977480. Archived from the original (PDF) on 2016-03-04.
4. ^ Dickson, Leonard Eugene (1920), "Rational solutions of x = y", History of the Theory of Numbers, vol. II, Washington, p. 687{{citation}}: CS1 maint: location missing publisher (link)
5. van Hengel, Johann (1888). "Beweis des Satzes, dass unter allen reellen positiven ganzen Zahlen nur das Zahlenpaar 4 und 2 für a und b der Gleichung a = b genügt". Pr. Gymn. Emmerich. JFM 20.0164.05.
6. Gleason, A. M.; Greenwood, R. E.; Kelly, L. M. (1980), "The twenty-first William Lowell Putnam mathematical competition (December 3, 1960), afternoon session, problem 1", The William Lowell Putnam mathematical competition problems and solutions: 1938-1964, MAA, p. 59, ISBN 0-88385-428-7
7. "21st Putnam 1960. Problem B1". 20 Oct 1999. Archived from the original on 2008-03-30.{{cite web}}: CS1 maint: bot: original URL status unknown (link)
8. Hausner, Alvin (November 1961). "Algebraic Number Fields and the Diophantine Equation m = n". The American Mathematical Monthly. 68 (9): 856–861. doi:10.1080/00029890.1961.11989781. ISSN 0002-9890.

External links

• "Rational Solutions to x^y = y^x". CTK Wiki Math. Archived from the original on 2021-08-15. Retrieved 2016-04-14.
• "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)

• Diophantine equations
• Recreational mathematics