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

Abstract

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.

Keywords

• diophantine tuples
• system of pellian equations
• linear forms in logarithms
• primary 11d09
• secondary 11d45, 11b37, 11j86