turkmath.org

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 (it was the set 1/16, 33/16, 17/4, 105/16). It is known that there are infinitely many Diophantine quadruples in integers (the first example, the set 1,3,8,120, was found by Fermat), and He, Togbe and Ziegler proved recently that there are no Diophantine quintuples in integers. Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple. It is still an open question whether there exist any rational Diophantine septuple. In this talk, we describe several constructions of infinite families of rational Diophantine sextuples. These constructions use properties of corresponding elliptic curves. We will also mention some other connections between Diophantine tuples and elliptic curves, including construction of high-rank elliptic curves over Q with given torsion group.