Jeudi 17 Mars 2016 (15h00, Salle G001)

Sadek Bouroubi, USTHB, Alger, Algérie.

Titre : Sur les m-uples Diophantiens et les partitions d'un entier.

Résumé :  The Greek mathematician Diophantus of Alexandria first studied the problem of finding four numbers such that the product of any two of them increased by unity is a perfect square. He found {1/16; 33/16; 17/4; 105/16} , a set of four positive rationals verifying this property. However, the first set of four positive integers with the above property, {1; 3; 8; 120} , was found by Fermat. Euler found the infinite family of such sets, {a; b; a + b + 2r; 4r(r + a)(r +b)g} , where ab + 1 = r^2 . He was also able to add the fifth positive rational ,777480/8288641 , to the Fermat's set. In January 1999, the first example of a set of six positive rationals with the property of Diophantus was found by Gibbs {11/192; 35/192; 155/27; 512/27; 1235/48; 180873/16} . These examples motivate the following definitions:

Definition 1 A set of m positive integers {a1; a2; ... ; am} is called a Diophantine m-tuple if aiaj + 1 is a perfect square for all 1  i \leq  j \leq  m.

It is natural to ask how large these sets, can be?

This question will be discussed and some results linked to integer

partitions will be presented.