On the family of \(D(4)\)-triples \(\{k-2, k+2, 4k^3-4k\}\) (Q2636785)

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 a special Diophantine \(D(4)\)-triples to a \(D(4)\)-quadruples. In particular, they prove that the quadruple \[ (k-2,k+2,4k^3-4k,d) \] is a Diophantine \(D(4)\)-quadruple only if \(d=4k\) or \(d=4k^5-12k^3+8k\), provided \(k\geq 3\).