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In general,
exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...
fails to be
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
. However, the
equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...
x^y = y^x has an infinity of solutions, consisting of the line and a
smooth curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
intersecting the line at , where is
Euler's number The number is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can ...
. The only integer solution that is on the curve is .


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 number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s are (2, 4) and (4, 2), although there are infinitely many solutions in
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s, such as (\tfrac, \tfrac) and (\tfrac, \tfrac). 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 Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
. J. van Hengel pointed out that if r, n are positive
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s with r \geq 3, then r^ > (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, which prompted Alvin Hausner to extend results to
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s.


Positive real solutions

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Explicit form

An infinite set of trivial solutions in positive
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s 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^ = c' \Rightarrow a' = W(c') to isolate the desired variable. :\begin y^x &= x^y = \exp\left(y\ln x\right) & \\ y^x \exp\left(-y\ln x\right) &= 1 & \left(\mbox \exp\left(-y\ln x\right)\right) \\ y\exp\left(-y\frac\right) &= 1 & \left(\mbox 1/x\right) \\ -y\frac\exp\left(-y\frac\right) &= \frac & \left(\mbox \frac\right) \end :\Rightarrow -y\frac = W\left(\frac\right) :\Rightarrow y = \frac\cdot W\left(\frac\right) = \exp\left(-W\left(\frac\right)\right) Where in the last step we used the
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), an ...
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 W_0\left(\frac\right) &= -\ln x \quad&\text &0 < x \le e, \\ W_\left(\frac\right) &= -\ln x \quad&\text &x \ge e. \end * 0 < x \le 1: :\Rightarrow \frac \ge 0 :\begin\Rightarrow y &= \exp\left(-W_0\left(\frac\right)\right) \\ &= \exp\left(-(-\ln x)\right) \\ &= x \end * 1 < x < e: :\Rightarrow \frac < \frac < 0 :\Rightarrow y = \begin \exp\left(-W_0\left(\frac\right)\right) = x \\ \exp\left(-W_\left(\frac\right)\right) \end *x = e: :\Rightarrow \frac = \frac :\Rightarrow y = \begin \exp\left(-W_0\left(\frac\right)\right) = x \\ \exp\left(-W_\left(\frac\right)\right) = x \end * x > e: :\Rightarrow \frac < \frac < 0 :\Rightarrow y = \begin \exp\left(-W_0\left(\frac\right)\right) \\ \exp\left(-W_\left(\frac\right)\right) = x \end 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^ = (x^v)^x. Raising both sides to the power \tfrac and dividing by x, we get : v = x^. Then nontrivial solutions in positive real numbers are expressed as the
parametric equation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters. In the case ...
The full solution thus is (y=x) \cup \left(v^,v^\right) \text v > 0, v \neq 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: :\frac = v^2\left(\frac\right) for v > 0 and v \neq 1. Setting v=2 or v=\tfrac generates the nontrivial solution in positive integers, 4^2=2^4. Other pairs consisting of
algebraic number In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...
s exist, such as \sqrt 3 and 3\sqrt 3, as well as \sqrt and 4\sqrt . 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\neq 1. For example, x^8=y has a slope of 8^2 at (\sqrt \sqrt 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\infty, so : x = \lim_v^ = \lim_\left(1+\frac 1n\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 \infty, the nontrivial solution asymptotes to the line y = 1. A more complete asymptotic form is : y = 1 + \frac + \frac \frac + \cdots.


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=\frac, y=\frac 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 = \sqrt produces a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
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^ \quad \mathrm \quad 0 y=e^ \quad \mathrm \quad 1/e 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 \sqrt /math> 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''.


References


External links

* * * * {{OEIS el, sequencenumber=A073084, name=Decimal expansion of −x, where x is the negative solution to the equation 2^x {{= x^2 Diophantine equations Recreational mathematics