Positive Rational Solutions to xy = ymx: A Number-Theoretic Excursion

1. PROLOGUE. Late in the last millennium, the second author ran a seminar course for undergraduates that was intended to introduce them to problem-solving and question-asking in the context of mathematical research. He led them through the classic " difficult " equation x y = y x (x, y > 0), (1) whose solution is much easier than one would think at first glance. Solutions were sought, first over R, then over Z, and finally over Q. In the context of this seminar, it was natural to consider the variant equation x y = y 2x (x, y > 0), (2) which does not seem to have appeared in the literature. It turns out that there are solutions to (2) that do not fit the well-known parametric pattern of (1) (see (5)). For example, x = 4 5 128 , y = 4 5 125 (3) is a solution to (2). This preposterous fact is trivial to verify: simply substitute (3) into (2), take logs, and transpose: 2x y = 2 4 5 3 = 128 125 = log x log y. As we shall see, the ultimate explanation for this identity is that 2 · 4 3 = 5 3 + 3. Upon discovering (3), the second author realized he was in over his head and contacted the first author. This paper is the result. 2. x y = y x. First, we review the familiar, but beautiful solution to (1), reserving historical references to the last paragraph of the section. We acknowledge the solutions x = y and now let y = t x, where t = 1, so that x y = y x ⇐⇒ x t x = (t x) x ⇐⇒ x t = t x ⇐⇒ x = t 1 t−1. (4) The positive real solutions to (1) are thus (x, y) = (u, u), (x, y) = (t 1 t−1 , t t t−1). (We might equally well have set y = x r in (1) and drawn essentially the same conclusion .) Since (1) implies that x 1/x = y 1/y and since f (u) = u 1/u increases on (1, e) and decreases on (e, ∞), for each x in (1, e) there is exactly one y in (e, ∞) so that (1) January 2004] POSITIVE RATIONAL SOLUTIONS TO x y = y mx 13