Pell's equation and Fermat

Abstract

Given that it took until the 19th century for the case n = 5 of Fermat's last theorem to be settled, it is not surprising that Fermat's claim of having a proof for all exponents greater than 2 is nowadays treated with considerable scepticism. However, perhaps the most important aspect of his claim has been the impetus it has given to the development of mathematical techniques over the three and a half centuries leading up to the proof by Wiles and Taylor [1, 2]. Not least amongst these techniques has been Fermat's own idea of proof by descente infinie [3,4].