60 research outputs found
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
We exhibit several families of elliptic curves with torsion group isomorphic
to and generic rank at least . 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 and rank , which is the
current record
- …