The Pell equation, by Edward Everett Whitford.

58

THE PELL EQUATION

The solution of this challenge problem filled Frenicle
and Fermat with great respect for Brouncker and Wallis.1
Fermat appears at one time to have been fully satisfied with the solution,2 but later he points out that
although Brouncker and Wallis had given many particular solutions they had not supplied a general proof,3
meaning presumably a proof that the solution is always
possible and that the method will always lead to the
solution sought.4
Fermat nowhere publishes his solution. It seems very
probable that it was much the same as Lord Brouncker's,
and that if he had found a better solution he would have
made it known. In fact Fermat is sometimes credited
with Brouncker's solution, due to a misunderstanding
of a sentence in Ozanam's Algebra, which in speaking
of the Brouncker-Wallis solution, reads: "Une regle
generale pour resoudre sette question, qui est de M. de
Fermat." The ambiguity as to what the pronoun
referred may have led Lagrange and others to suppose
the rule came from Fermat.5
Without having seen the solution of Wallis and
Brouncker, Malebranche, in 1658 or a little later, wrote
an article upon the equation.6 He was always able to
find a solution when the difference of the number given
1, Commercium epistolicum," XXXVIII, p. 846. "Oeuvres de
Fermat," vol. III, p. 577.
2 "Oeuvres de Fermat," vol. II, p. 402.
3 "Oeuvres de Fermat," vol. II, p. 433. See also as to the contradictory opinions of Frenicle and an anonymous Latin writer, "Oeuvres de
Fermat," vol. III, p. 592 and p. 605. In the last passage it reads, "Our
analysts do not recognize a vestige of proof there."
4 T. L. Heath, "Diophantus," 2d ed., p. 287.
5 T. L. Heath, "Diophantus," 2d ed., p. 288.
6C. Henry, "Recherches sur les manuscrits de Pierre de Fermat
suivres de fragments inedits de Bachet et de Malebranche," Bullettino di
bibliografia e di storia delle scienze matematiche e fisiche, vol. XII, p.
696, Rome, 1879. This MS. is preserved in the Bibliotheque Nationale
in Paris and is entitled "Essai de resolution par Malebranche de l'equation
Ax2 + 1 = y2," Fonds Frangais, No. 25308, p. 9-56.