Counterexamples to Dujella's conjecture on integral points on the elliptic curve attached to a Diophantine triple

Abstract

A set $\{a,b,c\}$ of distinct positive integers is called a Diophantine triple if $ab+1$, $ac+1$, and $bc+1$ are perfect squares. Dujella formulated the conjecture that the only integral $x$-coordinates on the attached elliptic curve $$y^2=(ax+1)(bx+1)(cx+1)$$ are $0,d_-,d_+$, together with $-1$ when the triple contains $1$, where $d_{\pm}=a+b+c+2abc \pm2\sqrt{(ab+1)(ac+1)(bc+1)}$. The weaker question was stated as Problem 4.8 in Dujella's list of open problems: must every integral point with $x\ne-1$ make $ax+1$, $bx+1$, and $cx+1$ all perfect squares? We construct infinitely many triples $\{5,115,c\}$ admitting an integral point with $x\ne-1$ for which all three factors are nonsquares.