Perlen-Kolloquium: Michel Waldschmidt (Paris 6)

On the so-called Fermat–Pell Equation x^2−dy^2 =±1

The equation x²−dy² = ±1, where the unknowns x and y are positive integers while d is a fixed positive integer which is not a square, has been mistakenly called with the name of Pell by Euler. It was investigated by Indian mathematicians since Brahmagupta (628) who solved the case d = 92, next by Bhaskara II (1150) for d = 61 and Narayana (during the 14-th Century) for d = 103. The smallest solution of x² − dy² = 1 for these values of d are respectively

11512 −92·1202 = 1, 17663190492 −61·2261539802 = 1

and

2275282 −103·224192 = 1,

hence they have not been found by a brute force search!
After a short introduction to this long history we explain the connection with Diophantine approximation and continued fractions, next we say a few words on more recent developments of the subject.

Veranstaltung übernehmen als iCal