Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form where n is a given positive nonsquare integer, and integer solutions are sought for x and y. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever the curve passes through a point whose x and y coordinates are both integers, such as the trivial solution with x = 1 and y = 0. Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y.
This equation was first studied extensively in India starting with Brahmagupta,[1] who found an integer solution to in his Brāhmasphuṭasiddhānta circa 628.[2] Bhaskara II in the 12th century and Narayana Pandit in the 14th century both found general solutions to Pell's equation and other quadratic indeterminate equations. Bhaskara II is generally credited with developing the chakravala method, building on the work of Jayadeva and Brahmagupta. Solutions to specific examples of Pell's equation, such as the Pell numbers arising from the equation with n = 2, had been known for much longer, since the time of Pythagoras in Greece and a similar date in India. William Brouncker was the first European to solve Pell's equation. The name of Pell's equation arose from Leonhard Euler mistakenly attributing Brouncker's solution of the equation to John Pell.[3][4][note 1]
- ^ O'Connor, J. J.; Robertson, E. F. (February 2002). "Pell's Equation". School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 13 July 2020.
- ^ Dunham, William. "Number theory – Number theory in the East". Encyclopedia Britannica. Retrieved 4 January 2020.
- ^ As early as 1732–1733 Euler believed that John Pell had developed a method to solve Pell's equation, even though Euler knew that Wallis had developed a method to solve it (although William Brouncker had actually done most of the work):
- Euler, Leonhard (1732–1733). "De solutione problematum Diophantaeorum per numeros integros" [On the solution of Diophantine problems by integers]. Commentarii Academiae Scientiarum Imperialis Petropolitanae (Memoirs of the Imperial Academy of Sciences at St. Petersburg) (in Latin). 6: 175–188. From p. 182: "At si a huiusmodi fuerit numerus, qui nullo modo ad illas formulas potest reduci, peculiaris ad invenienda p et q adhibenda est methodus, qua olim iam usi sunt Pellius et Fermatius." (But if such an a be a number that can be reduced in no way to these formulas, the specific method for finding p and q is applied which Pell and Fermat have used for some time now.) From p. 183: "§. 19. Methodus haec extat descripta in operibus Wallisii, et hanc ob rem eam hic fusius non-expono." (§ 19. This method exists described in the works of Wallis, and for this reason I do not present it here in more detail.)
- Lettre IX. Euler à Goldbach, dated 10 August 1750 in: Fuss, P. H., ed. (1843). Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siècle ... [Mathematical and physical correspondence of some famous geometers of the 18th century ...] (in French, Latin, and German). St. Petersburg, Russia. p. 37. From page 37: "Pro hujusmodi quaestionibus solvendis excogitavit D. Pell Anglus peculiarem methodum in Wallisii operibus expositam." (For solving such questions, the Englishman Dr. Pell devised a singular method [which is] shown in Wallis' works.)
- Euler, Leonhard (1771). Vollständige Anleitung zur Algebra, II. Theil [Complete Introduction to Algebra, Part 2] (in German). Kayserlichen Akademie der Wissenschaften (Imperial Academy of Sciences): St. Petersburg, Russia. p. 227. From p. 227: "§98. Hierzu hat vormals ein gelehrter Engländer, Namens Pell, eine ganz sinnreiche Methode erfunden, welche wir hier erklären wollen." (§ 98 Concerning this, a learned Englishman by the name of Pell has previously found a quite ingenious method, which we will explain here.)
- English translation: Euler, Leonhard (1810). Elements of Algebra ... Vol. 2 (2nd ed.). London, England: J. Johnson. p. 78.
- Heath, Thomas L. (1910). Diophantus of Alexandria : A Study in the History of Greek Algebra. Cambridge, England: Cambridge University Press. p. 286. See especially footnote 4.
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