THE UNIQUE EXENSIONS OF TWO PARAMETRIC FAMILIES OF DIOPHANTINE TRIPLES (Problems and prospects in Analytic Number Theory)

Abstract

This note gives a summary of the paper [M. Cipu, Y. Fujita, M. Mignotte, Two-parameter families of uniquely extendable Diophantine triples, to appear in Sci. China Math.]. For a nonzero integer n, a set of rn positive integers is called a D(n)-m-tuple if the product of any two distinct elements increased by n is a perfect square. Let A, K be positive integers and ε ∈ {-2, -1, 1, 2}. The main theorem of this note asserts that each of the D(ε²)-triples {K, A²K + 2εA, (A+ 1)² K + 2ε(A + l)} has unique extension to a D(ε²)-quadruple.