Diophantine triples of Fibonacci numbers (Q2833583)

An \(m\)-tupel \((a_1,\dots,a_m)\) of positive distinct integers is called a Diophantine \(m\)-tuple if \(a_ia_j+1\) is a perfect square for all \(1\leq i<j\leq m\). Let us denote by \(F_n\) the \(n\)-th Fibonacci number. In [Proc. Am. Math. Soc. 127, No. 7, 1999--2005 (1999; Zbl 0937.11011)] \textit{A. Dujella} proved that if \((F_{2n},F_{2n+2},F_{2n+4},d)\) is a Diophantine quadruple, then \(d=4F_{2n+1}F_{2n+2}F_{2n+3}\). In the paper under review the previous result is further extended by showing that if \((F_{2n},F_{2n+2},F_k)\) is a Diophantine triple then \(k\in\{2n+4,2n-2\}\) provided that \(n>2\). The authors also state the conjectures that no Diophantine quadruple consisting of Fibonacci numbers exists and that if \((F_p,F_q,F_r)\) is a Diophantine triple with \(p<q<r\), then \((p,q,r)=(2n,2n+2,2n+4)\) or \((1,4,6)\).