Equation xy = yx explained

x ^{y} = y ^{x} has solutions, such as x =2 , y =4 .

History

The equation

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

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.

The problem was discussed in a number of publications. In 1960, the equation was among the questions on the William Lowell Putnam Competition,^[1] which prompted Alvin Hausner to extend results to algebraic number fields.^[2]

Positive real solutions

Main source:

Explicit form

An infinite set of trivial solutions in positive real numbers is given by

x = y. Nontrivial solutions can be written explicitly using the Lambert W function. The idea is to write the equation as ae ^{b} = c and try to match a and b by multiplying and raising both sides by the same value. Then apply the definition of the Lambert W function a'e ^{a'} = c' ⇒ a' = W(c') to isolate the desired variable.\begin{align} y ^{x} & = x ^{y} = \exp\left(y ln x\right) & \\ y ^{x} \exp\left( - y ln x\right) & = 1 & \left( multiplyby \exp\left( - y ln x\right)\right) \\ y\exp\left( - y (ln x)/(x) \right) & = 1 & \left( raiseby 1 /x\right) \\ - y (ln x)/(\exp\left( - y x) (ln x)/(x) \right) & = (-ln x)/(x) & \left( multiplyby (-ln x)/(x) \right) \end{align} ⇒ - y (ln x)/(x) = W\left( (-ln x)/(x) \right) ⇒ y = (- x)/(ln x) ⋅ W\left( (-ln x)/(x) \right) = \exp\left( - W\left( (-ln x)/(x) \right)\right) 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:

\begin{align} W _{0 \left( (-ln x)/(x)} \right) & = -ln x & for & 0 < x \le e, \\ W _{-1} \left( (-ln x)/(x) \right) & = -ln x & for &x \ge e. \end{align} 0 < x \le 1 :⇒ (-ln x)/(x) \ge 0 \begin{align} ⇒ y & = \exp\left( - W _{0 \left( (-ln x)/(x)} \right)\right) \\ & = \exp\left( - ( -ln x)\right) \\ & = x \end{align} 1 < x < e :⇒ (-1)/(e) < (-ln x)/(x) < 0 ⇒ y = \begin{cases} \exp\left( - W _{0 \left( (-ln x)/(x)} \right)\right) = x \\ \exp\left( - W _{-1} \left( (-ln x)/(x) \right)\right) \end{cases} x = e ⇒ (-ln x)/(x) = (-1)/(e) ⇒ y = \begin{cases} \exp\left( - W _{0 \left( (-ln x)/(x)} \right)\right) = x \\ \exp\left( - W _{-1} \left( (-ln x)/(x) \right)\right) = x \end{cases} x > e :⇒ (-1)/(e) < (-ln x)/(x) < 0 ⇒ y = \begin{cases} \exp\left( - W _{0 \left( (-ln x)/(x)} \right)\right) \\ \exp\left( - W _{-1} \left( (-ln x)/(x) \right)\right) = x \end{cases}

Hence the non-trivial solutions are:

Parametric form

Nontrivial solutions can be more easily found by assuming

x \ne 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 , we getv = x ^{v-1} . Then nontrivial solutions in positive real numbers are expressed as the parametric equation

The full solution thus is

(y = x) \cup \left(v ^{1/(v-1)} ,v ^{v/(v-1)} \right) for v > 0 , v ≠ 1 .

Based on the above solution, the derivative

dy/dx is 1 for the (x,y) pairs on the line y = x, and for the other (x,y) pairs can be found by (dy/dv)/(dx/dv), which straightforward calculus gives as:(dy)/(dx) = v ^{2 \left( (v -1-ln v)/(v -1- v ln v)} \right) for v > 0 and 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 parameterization above leads to a geometric property of this curve. It can be shown that

x ^{y} = y ^{x} describes the isocline curve where power functions of the form x ^{v} have slope v ^{2} for some positive real choice of v ≠ 1 . For example, x ^{8=y} has a slope of 8 ^{2} at (\sqrt[ 7 ]{ 8 }, \sqrt[ 7 ]{ 8 } ^{8),} which is also a point on the curve x ^{y=y} ^{x.}

The trivial and non-trivial solutions intersect when

v = 1 . The equations above cannot be evaluated directly at v = 1 , 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 inf ty , sox = \lim _{v\to} v ^{1/(v-1)} = \lim _{n\toinfty} \left( 1+ (1 n\right))/(^{n}) = e. Thus, the line y = x and the curve for x ^{y-y} ^{x} = 0 , y \ne x intersect at .

As

x \to inf ty , the nontrivial solution asymptotes to the line y = 1 . A more complete asymptotic form isy = 1 + (ln x)/(x) + (3)/(2) (( ln x) ^{2})/(x ^{2}) + … .

Other real solutions

An infinite set of discrete real solutions with at least one of

x and y negative also exist. These are provided by the above parameterization when the values generated are real. For example, x = (1)/(\sqrt[ 3 ]{ -2) }, y = (-2)/(\sqrt[ 3 ]{ -2) } is a solution (using the real cube root of -2 ). Similarly an infinite set of discrete solutions is given by the trivial solution y = x for x< 0 when x ^{x} is real; for example x = y =-1 .

Similar graphs

Equation

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 to infinity.

The curved section can be written explicitly as

y = e ^{W ^{x))} _{0 ( ln (x}} for 0 <x< 1 /e, y = e ^{W _{-1} ( ln (x ^{x))}} for 1 /e<x< 1 .

This equation describes the isocline curve where power functions have slope 1, analogous to the geometric property of

x ^{y} = y ^{x} described above.

The equation is equivalent to

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

Equation

The equation

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.

External links

• Web site: Rational Solutions to x^y y^x . CTK Wiki Math . 2016-04-14 . 2021-08-15 . https://web.archive.org/web/20210815091140/https://www.cut-the-knot.org/wiki-math/index.php?n=Algebra.RationalSolutionOfXYYX . dead .
• Web site: x^y y^x - commuting powers . Torsten Sillke . Arithmetical and Analytical Puzzles . https://web.archive.org/web/20151228091303/https://www.math.uni-bielefeld.de/~sillke/PUZZLES/x%5Ey-x%5Ey . 2015-12-28.
• Web site: Parametric Graph of x^yy^x . . dborkovitz . 2012-01-29.

Notes and References

1. Web site: 21st Putnam 1960. Problem B1 . 20 Oct 1999 . bot: unknown . https://web.archive.org/web/20080330183949/http://www.kalva.demon.co.uk/putnam/putn60.html . 2008-03-30 .
2. Hausner . Alvin . November 1961 . Algebraic Number Fields and the Diophantine Equation mⁿ = n^m . . 68 . 9 . 856–861 . 10.1080/00029890.1961.11989781 . 0002-9890.