The relative upper bound for the third element in a \(D(-1)\)-quadruple (Q2915716)

A \(D(n)\)-\(m\)-tuple is an \(m\)-tuple \((a_1,\dots,a_m)\) of positive integers such that for all \(1\leq i<j\leq m\) \(a_ia_j+n\) is a perfect square. The paper under review is concerned with the case that \(n=-1\). In this case it is a folklore conjecture that no \(D(-1)\)-quadruple exists. Due to \textit{A. Dujella} and \textit{C. Fuchs} [J. Lond. Math. Soc. (2) 71, No. 1, 33--52 (2005; Zbl 1166.11309)] it is known that at least no \(D(-1)\)-quintuple exists and that a \(D(-1)\)-quadruple \((a,b,c,d)\) with \(a<b<c<d\) satisfies \(a=1\).NEWLINENEWLINEIn the paper under review the authors improve their previously found upper bound [\textit{A. Filipin} and \textit{Y. Fujita}, Math. Commun. 15, No. 2, 387--391 (2010; Zbl 1213.11067)] for \(c\) in terms of \(b\). In particular, they show that if \((1,b,c,d)\) is a \(D(-1)\)-quadruple with \(b<c<d\), then \(c<9.6 b^4\).