The extensibility of the Diophantine triple {2, b, c}

The aim of this paper is to consider the extensibility of the Diophantine triple {2, b, c}, where 2 < b < c, and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of c’s (depending on b). As corollary, for example, we prove that for b/2 − 1 prime, all Diophantine quadruples {2, b, c, d} with 2 < b < c < d are regular.

High rank elliptic curves induced by rational Diophantine triples

A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over ℚ with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank ≥ 7, which are both the current records for that kind of curves.

On the number of extensions of a Diophantine triple

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by unity is a perfect square. Any Diophantine triple is conjectured to be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element in the triple. A previous work of the second and third authors revealed that the number of such extensions for a fixed Diophantine triple is at most 11. In this paper, we show that the number is at most eight.

On the extendibility of the Diophantine triple{1,5,c}

We study the problem of extendibility of the triples of the form{1,5,c}. We prove that ifck=sk2+1, where(sk)is a binary recursive sequence,kis a positive integer, and the statement that all solutions of a system of simultaneous Pellian equationsz2−ckx2=ck−1,5z2−cky2=ck−5are given by(x,y,z)=(0,±2,±sk), is valid for2≤k≤31, then it is valid for all positive integerk.