Equation xʸ=yˣ

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 ≠ y , the only solutions in natural numbers are ( 2 , 4 ) and ( 4 , 2 ) , although there are infinitely many solutions in rational numbers. The reply by Goldbach (31 January 1729) contains general solution of the equation obtained by substituting y = v x . A similar solution was found by Euler.

J. van Hengel pointed out that if r , n are positive integers, r ⩾ 3 or n ⩾ 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 questions on William Lowell Putnam Competition which prompted A. Hausner to extend results to algebraic number fields.

Positive real solutions

An infinite set of trivial solutions in positive real numbers is given by x = y .

Nontrivial solutions can be found by assuming x ≠ y and letting y = v x . Then

Raising both sides to the power 1 x and dividing by x ,

Then nontrivial solutions in positive real numbers are expressed as

Setting v = 2 or v = 1 2 generates the nontrivial solution in positive integers, 4 2 = 2 4 .