The problem of Diophantus and Davenport

In this paper we describe the author\u27s results concerning the problem of the existence of a set of four or five positive integers with the property that the product of its any two distinct elements increased by a fixed integer n is a perfect square

Elliptic curves with torsion group Z/6Z\Z /6\Z

We exhibit several families of elliptic curves with torsion group isomorphic to $\Z/6\Z$ and generic rank at least $3$. Families of this kind have been constructed previously by several authors: Lecacheux, Kihara, Eroshkin and Woo. We mention the details of some of them and we add other examples developed more recently by Dujella and Peral, and MacLeod. Then we apply an algorithm of Gusi\'c and Tadi\'c and we find the exact rank over \Q(t) to be 3 and we also determine free generators of the Mordell-Weil group for each family. By suitable specializations, we obtain the known and new examples of curves over \Q with torsion $\Z/6\Z$ and rank $8$, which is the current record

The problem of Diophantus and Davenport

In this paper we describe the author\u27s results concerning the problem of the existence of a set of four or five positive integers with the property that the product of its any two distinct elements increased by a fixed integer n is a perfect square