Solving the Pell Equation, Volume 49, Number 2

to be solved in positive integers x , y for a given nonzero integer d. For example, for d = 5 one can take x = 9, y = 4. We shall always assume that d is positive but not a square, since otherwise there are clearly no solutions. The English mathematician John Pell (1610– 1685) has nothing to do with the equation. Euler (1707–1783) mistakenly attributed to Pell a solution method that had in fact been found by another English mathematician, William Brouncker (1620–1684), in response to a challenge by Fermat (1601–1665); but attempts to change the terminology introduced by Euler have always proved futile. Pell’s equation has an extraordinarily rich history, to which Weil’s book [13] is the best guide; see also [3, Chap. XII]. Brouncker’s method is in substance identical to a method that was known to Indian mathematicians at least six centuries earlier. As we shall see, the equation also occurred in Greek mathematics, but no convincing evidence that the Greeks could solve the equation has ever emerged. A particularly lucid exposition of the “Indian” or “English” method of solving the Pell equation is found in Euler’s Algebra [4, Abschn. 2, Cap. 7]. Modern textbooks usually give a formulation in terms of continued fractions, which is also due to Euler (see for example [9, Chap. 7]). Euler, as well as his Indian and English predecessors, appears to take it for granted that the method always produces a solution. That is true, but it is not obvious—all that is obvious is that if there is a solution, the method will find one. Fermat was probably in possession of a proof that there is a solution for every d (see [13, Chap. II, § XIII]), and the first to publish such a proof was Lagrange (1736–1813) in 1768 (see Figure 1). One may rewrite Pell’s equation as