On a family of two-parametric \(D(4)\)-triples (Q2901982)

A Diophantine \(D(n)\)-\(m\)-tuple is an \(m\)-tuple of distinct positive integers \((a_1,\ldots,a_m)\) such that \(a_ia_j+n=\square\) for all \(1\leq i <j\leq m\). In the paper under review the authors prove results on the possibilities of extending special Diophantine \(D(4)\)-triples to \(D(4)\)-quadruples. In particular, let \(A\) and \(k\) be integers and assume that \(1\leq A \leq 22\) or \(A\geq 51767\). Then they prove that the quadruple NEWLINE\[NEWLINE(k,A^2k+4A,(A+1)^2k+4(A+1),d)NEWLINE\]NEWLINE is a Diophantine \(D(4)\)-quadruple if and only if NEWLINE\[NEWLINE (A^4+2A^3+A^2)k^3+(8A^3+12A^2+4A)k^2+(20A^2+20A+4)k+(16A+8). NEWLINE\]