# Fermat et l'équation de Pell

Does any of you happen to own and electronic copy of the paper "Fermat et l'équation de Pell" by A. Weil?

If I understand correctly, this paper can be found on pages 413-419 of the third volume of Weil's Oeuvres Scientifiques.

In case you have it at hand and have read it already, would you be so kind as to briefly tell us the main points Weil made therein regarding Fermat's actual contributions to the theory of the Pell equation?

• I am not a mathematician but searching the title in Google Books shows that the chapter is (also?) present in "Prismata: Naturwissenschaftsgeschichtliche Studien. Festschrift für Willy Hartner , Edited by. Y. Maeyama , W. G. Saltzer", 1977. Prismata is written in Greek letters in the book title. May 2, 2022 at 3:30
• Michael Sean Mahoney, "The Mathematical Career of Pierre de Fermat 1601-1665, 2nd ed." p. 328: In "Fermat et l'équation de Pell," Weil offers a reconstruction suggested by Bachet's integral solution of the linear equation $Ax - By = C$ and by Euler's description of "eine sehr sinnreiche Methode" that he attributed to "ein gelehrter Engländer namens Pell". That's all Google snippets provide. May 2, 2022 at 6:26

In this short article he explains the history of the solution of Pell's equation. It was proposed by Fermat in his letter to British mathematicians, and solved by Wallis and Brouncker, jointly, in their correspondence. They explained their solution on a number of special cases. Fermat criticized this solution asserting that they did not give a general proof. Then Weil speculates whether Fermat himself had a "general proof" himself and what this proof could be. He also analyses the "British solution", and establishes its connection with the Euclidean algorithm and a continued fraction algorithm.

The first published complete proof is due to Lagrange. Weil also gives some detail about pre-history of this solution, and mentions Bachet and Pascal. In the beginning he notices that there is no known connection of this equation with John Pell, who was famous as a mathematician during his life time but published nothing.

• Thanks a bunch for your reply... Does he mention the chakravala method in the paper? May 2, 2022 at 15:36
• @José Hdz. Stgo: no, he does not. Neither he mentions Archimedes or other Greeks. He is strictly limited to Fermat's contemporaries. May 2, 2022 at 21:54