Equation xʸ=yˣ

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

History

The equation $x^{y}=y^{x}$ 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 $(2,4)$ and $(4,2)$ , 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 $y=vx$ .^[3] A similar solution was found by Euler.^[4]

J. van Hengel pointed out that if $r,n$ are positive integers, $r\geqslant 3$ or $n\geqslant 3$ then $r^{r+n}>(r+n)^{r}$ ; 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 $y=vx$ . Then

(vx)^{x}=x^{vx}=(x^{v})^{x}

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

v=x^{v-1}

Then nontrivial solutions in positive real numbers are expressed as

x=v^{\frac {1}{v-1}}

y=v^{\frac {v}{v-1}}

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

References

1 2 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.
1 2 3 David Singmaster. "Sources in recreational mathematics: an annotated bibliography. 8th preliminary edition". Archived from the original on April 16, 2004. CS1 maint: Unfit url (link)
1 2 3 4 Marta Sved (1990). "On the Rational Solutions of x^y = y^x" (PDF). Mathematics Magazine. Archived from the original (PDF) on 2016-03-04.
1 2 3 4 Leonard Eugene Dickson (1920), "Rational solutions of x^y = y^x", History of the Theory of Numbers, II, Washington, p. 687
↑ 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".
↑ 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
↑ "21st Putnam 1960. Problem B1". 20 Oct 1999.
↑ 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.

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