Abstract

Let k be a natural number. Two integers α and β are said to have the property D(k) (resp. D(-k)) if αβ+k (resp. αβ-k) is a perfect square. The purpose of this paper is identication of polynomials producing quadruples with property D(k) for certain values of k. Incidentally the paper brings out an at- tribute of Ramanujan number 1729 in contributing two quadruples of polynomials with property D(k).

Keywords and Phrases

Property pk, extendable set, Pr;k sequence, Pell's equation, quadruple with Diophantine property, Ramanujan number.

A.M.S. subject classification

11B37, 11B83, 11D09.

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