Pell's equation

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. Quotes Florian Cajori, the noted historian, summed up the matter in an extraordinarily suggestive manner: The perversity of fate has willed it that the equation y2 = nx2 + 1 should now be called Pell’s Problem, while in recognition of Brahmin scholarship it ought to be called the “Hindu Problem.” It is a problem that has exercised the highest faculties of some of our greatest modern analysts. Indian mathematical historians would like to call it the Brahmagupta–Bhaskara problem, keeping in mind that Bhaskar perfected Brahmagupta’s method of solution in the twelfth century; Bhaskara used “Chakravala”, or a cyclic process, to improve Brahmagupta’s method by doing away with the necessity of finding a trial solution. (Bhattacharyya 2011: 187). Bhattacharyya, R. K. ‘Brahmagupta: The Ancient Indian Mathematician’. In Ancient Indian Leaps into Mathematics, edited by B. S. Yadav and Man Mohan, pp. 185-192. Birkhäuser, 2011. quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189 ‘[…] his correspondence with Digby, and, through Digby, with the English mathematicians WALLIS and BROUCKNER occupies the next year and a half, from January 1657 to June 1658. It begins with a challenge to Wallis and Brouckner, but at the same time also to Frenicle, Schooten “and all others in Europe” to solve a few problems, with special emphasis upon what later became known (through a mistake of Euler’s) as “Pell’s equation”. What would have been Fermat’s astonishment if some missionary, just back from India, had told him that his problem had been successfully tackled there by native mathematicians almost six centuries earlier!’ André Weil his book Number Theory: An Approach Through History from Hammurapi to Legendre quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189 Weil, André. Number Theory: An Approach through History from Hammurapi to Legendre. Boston: Birkhäuser, 1984. ‘[…] the chakravala method anticipated the European methods by more than a thousand years. But, as we have seen, no European performances in the whole field of algebra at a time much later than Bhaskara’s, nay nearly up to our times, equalled the marvellous complexity and ingenuity of chakravala.’ C. O. Selenius quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189 Selenius, C. O. ‘Rationale of the Chakravala process of Jayadeva and Bhaskara II’. Historia Mathematica 2 (1975): pp. 167-184. This method is supreme above all praise; it is certainly the finest thing accomplished in number theory before Lagrange. Diese Methode ist über alles Lob erhaben; sie ist sicherlich das Feinste, was in der Zahlenlehre vor Lagrange geleistet wurde. On Bhaskara II's method Hankel, H.: Zur Geschichte der Mathematik im Altertum und Mittelalter, S. 202. Teubner, Leipzig (1874). External links Category:Equations