Pell's equation

By a mistake of Euler, the Diophantine equation y^2 - Ax^2 = 1 has been erroneously known as "Pell's equation"; but, in fact, the English mathematician John Pell (1611-1685) did no more than copy it down in his papers, from Fermat's letters of 1657 and 1658.

For an extensive historical account on "Pell's equation", see Sir Thomas L. Heath, Diophantus of Alexandria : A Study in the History of Greek Algebra (Dover Pub., New York, 1931-1963, 552 pages), Supplement, Section II: "Equation y^2 - Ax^2 = 1, pp. 277-292. Particularly in page 285, after a presentation of the history of the equation up to Fermat's time (including citations to Pythagoreans, Archimedes, Diaphanous, and the Indian solution), one can read that:

" ... Fermat rediscovered the problem and was the first to assert that the equation x^2 - Ay^2 = 1, where A is any integer not a square, always has an unlimited number of solutions in integers. His statement was made in a letter to FrÃ©nicle of February, 1657 (cf. Oeuvres de Fermat, II, pp.333-4). Fermat asks FrÃ©nicle for a general rule for finding, when any number not a square is given, squares which, when they are respectively multiplied by the given number and unity is added to the product, give squares. If, says Fermat, FrÃ©nicle cannot give a general rule, will he give the smallest value of y which will satisfy the equations 61y^2 + 1 = x^2 and 109y^2 + 1 = x^2 ? (Footnote 3: Fermat evidently chose these cases for their difficulty; the smallest values satisfying the first equation are y=226153980, x=1766319049, and the smallest values satisfying the second are y=15140424455100, x=158070671986249)." And, after a extensive quotation of Fermat's letter, in page 286, one can read that: "The challenge was taken up in England by William, Viscount Brouncker, first President of the Royal Society, and Wallis (Footnote 1: An excellent summary of the whole story is given in Wertheim's paper "Pierre Fermat's Streit mit John Wallis" in Abhandlungen zur Gesch. der Math., IX. Heft (Cantor-Festschrit), 1899, pp.557-576). See also H. Konen, Geschichte der Gleichung t^2-Du^2=1, Leipzig (S. Hirzel), 1901). At first, owing apparently to some misunderstanding, they thought that only rational, and not necessarily integral solutions were wanted, and found of course no difficulty in solving this easy problem. Fermat was, naturally, not satisfied with this solution, and Brouncker, attacking the problem again, finally succeeded in solving it. The method is set out in letters of Wallis (Footnote 2: Oeuvres de Fermat, III, pp.457-480, 490-503) of 17th December, 1657, and 30th January, 1658, and in chapter XCVIII of Wallis' Algebra; Euler also explains it fully in his Algebra (Footnote 3: Part II, chap. VII), wrongly attributing it to Pell (Footnote 4: This was the origin of the erroneous description of our equation as the "Pellian" equation. Hankel (in Zur Geschichte der Math. im Alterthum und Mittlelalter, p.203) supposed that the equation was so called because the solution was reproduced by Pell in an English translation (1668) by Thomas Brancker of Rahn's Algebra; but this is a misapprehension, as the so-called "Pellian" equation is not so much as mentioned in Pell's additions (Wertheim in Bibliotheca Mathematica, III, 1902, pp.124-6); Konen, pp.33-4 note). The attribution of the solution to Pell as a pure mistake of Euler's, probably due to a cursory reading by him of the second volume of Wallis' Opera where the solution of the equation ax^2 + 1 = y^2 is given as well as information as to Pell's work in indeterminate analysis. But Pell is not mentioned in connexion with the equation at all (EnestrÃ¶m in Bibliotheca Mathematica, III, 1902, p.206)."

For more information about "Pell's equation", see Harold M. Edwards, The Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Springer-Verlag, New York, 1977, 410 pages), pp. 25-33. Particularly in page 33 one can read that

"This problem of Fermat is now known as "Pell's equation" as a result of a mistake on the part of Euler. In some way, perhaps from a confused recollection of Wallis's Algebra, Euler gained the mistaken impression that Wallis attributed the method of solving the problem not to Brouncker but to Pell, a contemporary of Wallis who is frequently mentioned in Wallis's works but who appears to have had nothing to do with the solution of Fermat's problem. Euler mentions this mistaken impression as early as 1730, when he was only 23 years old, and it is included in his definitive Introduction to Algebra written around 1770. Euler was the most widely read mathematical writer of his time, and the method from that time on has been associated with the name of Pell and the problem that it solved --- that of finding all integer solutions of y^2 - Ax^2 = 1 when A is a given number not a square --- has been known ever since as "Pell's equation", despite the fact that it was Fermat who first indicated the importance of the problem and despite the fact that Pell had nothing whatever to do with it."

See also AndrÃ© Weil, Number Theory : An approach through history - From Hammurapi to Legendre (BirkhÃ¤user, Boston, 1984, xv+375 pages), in many different pages. In particular, at page 174, one can read that:

"Pell's name occurs frequently in Wallis's Algebra, but never in connection with the equation x^2 - Ny^2 = 1 to which his name, because of Euler's mistaken attribution, has remained attached; since its traditional designation as "Pell's equation" is unambiguous and convenient, we will go on using it, even though it is historically wrong."

Raul Nunes ( raul_nunes@uol.com.br )
NEST Nunes' Exposition of Scientific Truths
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