MSI Weekly Bulletin - Week starting Monday 15 January, 2007

A Diophantine $m$-tuple is a set of $m$-positive integers $\{a_1, a_2, \ldots, a_m\}$ such as the product of any two of them plus $1$ is a square. For example, $\{1,3,8,120\}$ is a Diophantine quadruple. There are infinitely many Diophantine quadruples. It is conjectured that there is no Diophantine quintuple but this has not been proved yet. In my talk, I will survey what is known about this subject along with variations of it with rational contents, or polynomial contents, or replacing the squares by larger powers, etc. The talk will be aimed at a general audience. BIO: http://www.matmor.unam.mx/personal/fluca.htm