There Are Infinitely Many Rational Diophantine Sextuples

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple. In this talk, we describe construction of innitely many rational Diophantine sextuples. This is joint work with Matija Kazalicki, Miljen Miki¢ and Marton Szikszai.