Equation xy = yx

is mentioned in a letter of Bernoulli to Goldbach (29 June 1728[2]).

although there are infinitely many solutions in rational numbers, such as

[3][4] The reply by Goldbach (31 January 1729[2]) contains a general solution of the equation, obtained by substituting

[3] A similar solution was found by Euler.

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

Nontrivial solutions can be written explicitly using the Lambert W function.

by multiplying and raising both sides by the same value.

Then apply the definition of the Lambert W function

Here we split the solution into the two branches of the Lambert W function and focus on each interval of interest, applying the identities: Hence the non-trivial solutions are:

Nontrivial solutions can be more easily found by assuming

, we get Then nontrivial solutions in positive real numbers are expressed as the parametric equation

generates the nontrivial solution in positive integers,

Other pairs consisting of algebraic numbers exist, such as

The parameterization above leads to a geometric property of this curve.

describes the isocline curve where power functions of the form

The trivial and non-trivial solutions intersect when

, the nontrivial solution asymptotes to the line

A more complete asymptotic form is An infinite set of discrete real solutions with at least one of

These are provided by the above parameterization when the values generated are real.

is a solution (using the real cube root of

produces a graph where the line and curve intersect at

The curve also terminates at (0, 1) and (1, 0), instead of continuing on to infinity.

The curved section can be written explicitly as

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

Equivalently, this can also be shown to demonstrate that the equation

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.