The extension of the $$D(-k)$$-pair $$\{k,k+1\}$$ to a quadruple

Abstract

Let \(n\ne 0\) be an integer. A set of m distinct positive integers \(\{a_1,a_2,\ldots ,a_m\}\) is called a D(n)-m-tuple if \(a_ia_j + n\) is a perfect square for all \(1\le i < j \le m\). Let k be a positive integer. In this paper, we prove that if \(\{k,k+1,c,d\}\) is a \(D(-k)\)-quadruple with \(c>1\), then \(d=1\). The proof relies not only on standard methods in this field (Baker’s linear forms in logarithms and the hypergeometric method), but also on some less typical elementary arguments dealing with recurrences, as well as a relatively new method for the determination of integral points on hyperelliptic curves.