Equation xʸ=yˣ

Graph of

x y = y x

.

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 with $r\geq 3$ 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 the questions on the William Lowell Putnam Competition^[6]^[7] which prompted Alvin 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, $4^{2}=2^{4}.$

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

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

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

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

Similar graphs

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

The equation $log_{x}(y)=log_{y}(x)$ produces a graph where the curve and line intersect at (1,1). The curve (which is actually the positive section of y=1/x) becomes asymptotic to 0, as opposed to 1.

References

1 2 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.
1 2 3 Singmaster, David. "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 Sved, Marta (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 Dickson, Leonard Eugene (1920), "Rational solutions of x^y = y^x", History of the Theory of Numbers, II, Washington, p. 687
↑ 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 = b^a genügt".
↑ 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
↑ "21st Putnam 1960. Problem B1". 20 Oct 1999. Archived from the original on 2008-03-30.
↑ Hausner, Alvin (November 1961). "Algebraic Number Fields and the Diophantine Equation mⁿ = n^m". 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.
"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)

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[loczy-1] 1 2 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.

[Singmaster-2] 1 2 3 Singmaster, David. "Sources in recreational mathematics: an annotated bibliography. 8th preliminary edition". Archived from the original on April 16, 2004.CS1 maint: Unfit url (link)

[Sved1990-3] 1 2 3 4 Sved, Marta (1990). "On the Rational Solutions of x^y = y^x" (PDF). Mathematics Magazine. Archived from the original (PDF) on 2016-03-04.

[Dickson-4] 1 2 3 4 Dickson, Leonard Eugene (1920), "Rational solutions of x^y = y^x", History of the Theory of Numbers, II, Washington, p. 687

[Hengel1888-5] 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 = b^a genügt".

[wlp-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.

[8] Hausner, Alvin (November 1961). "Algebraic Number Fields and the Diophantine Equation mⁿ = n^m". The American Mathematical Monthly. 68 (9): 856–861. doi:10.1080/00029890.1961.11989781. ISSN 0002-9890.