The non-extendibility of some parametric families of \(D(-1)\)-triples (Q2907037)

The authors study the positive solutions to the equation \(T^2-(r^2+1)S^2=r^2\) with \(r\) a positive integer. Call such a solution regular if it is associated to one of the fundamental solutions \((r,0)\), \(\bigl( r^2-r-1, \pm (r-1)\bigr)\). It is noticed that for \(r=2q^2\) (\(q\in \mathbb N\)) there are classes of solutions belonging to \((2q^3+q,\pm q)\).NEWLINENEWLINEAccording to the main result of the paper, if \((t,s)\) is a regular solution to the equation of interest then the system of Diophantine equations \(Y^2-(r^2+1)X^2=r^2\), \(Z^2-(s^2+1)X^2=s^2\) has only the trivial solutions \((x,y,z)=(0, \pm r, \pm s)\). Moreover, if \(r=2q^2\) (\(q\in \mathbb N\)) then the same is true for any positive solution \((t,s)\) associated to one of \((2q^3+q,\pm q)\). It is also shown that if \(r=p^k\) or \(r=2p^k\) for an odd prime \(p\) and \(k\in \mathbb N\), the equation of interest has only regular solutions, except in the case \(r=2p^k\) with \(k\) even, when there are precisely two more classes of solutions.NEWLINENEWLINEFrom these results it is concluded that for any odd prime \(p\) and positive integer \(k\), the \(D(-1)\)-pair \((1,r^2+1)\) cannot be extended to a \(D(-1)\)-quadruple if either \(r=p^k\), \(2p^k\) or \(r^2+1=p^k\), \(2p^k\).