Equation xʸ=yˣ

In general, exponentiation fails to be commutative. However, the equation ${\displaystyle x^{y}=y^{x}}$ holds in special cases, such as ${\displaystyle x=2,y=4}$.^[1]

History

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 $x\neq y$, the only solutions in natural numbers are ${\displaystyle (2,4)}$ and ${\displaystyle (4,2)}$, although there are infinitely many solutions in rational numbers.^[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 $r,n$ are positive integers, ${\displaystyle r\geqslant 3}$ or ${\displaystyle n\geqslant 3}$ then ${\displaystyle r^{r+n}>(r+n)^{r}}$; therefore it is enough to consider possibilities $x=1$ and $x = 2$ 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 questions on William Lowell Putnam Competition^[6][7] which prompted A. Hausner to extend results to algebraic number fields.^[3][8]

Positive real solutions

Main source:^[1]

An infinite set of trivial solutions in positive real numbers is given by $x=y$.

Nontrivial solutions can be found by assuming $x\neq y$ 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 $x$,

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

Then nontrivial solutions in positive real numbers are expressed as

${\displaystyle x=v^{\frac {1}{v-1}}}$,

${\displaystyle y=v^{\frac {v}{v-1}}}$.

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

References

1. Lajos Lóczi. "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. David Singmaster. "Sources in recreational mathematics: an annotated bibliography. 8th preliminary edition". Archived from the original on April 16, 2004.
3. Marta Sved (1990). "On the Rational Solutions of x^y = y^x" (PDF). Mathematics Magazine. Archived from the original (PDF) on 2016-03-04.
4. Leonard Eugene Dickson (1920), "Rational solutions of x^y = y^x", History of the Theory of Numbers, II, Washington, p. 687
5. Hengel, Johann van (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 = b^a genügt".
6. A. M. Gleason, R. E. Greenwood, L. M. Kelly (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.
8. A. Hausner, Algebraic number fields and the Diophantine equation mⁿ = n^m, Amer. Math. Monthly 68 (1961), 856—861.

External links

• "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.
• "Sloane's A073084 : Decimal expansion of -x, where x is the negative solution to the equation 2^x = x^2". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.