Pellian equations of special type (Q2063376)

A Diophantine \( m \)-tuple with property \( D(-1) \) or just a \( D(-1) \)-\( m \)-tuple in a commutative ring \( R \) is a set of \( m \) nonzero elements of \( R \) such that \( ab-1 \) is a square in \( R \) for any two distinct elements \( a \) and \( b \) in the set. In the paper under review, the authors consider the solvability of the Pellian equation \[ x^2-(d^2+1)y^2=-m, \] in the cases: \( d=n^k \), \( m=n^{2\ell-1} \), where \( k \) and \( \ell \) are positive integers, \( n \) is a composite positive integer; and \( d=pq \), \( m=pq^2 \), where \( p \) and \( q \) are primes. Furthermore, the authors use the results obtained to prove results on the extensibility of some \( D(-1) \)-pairs to quadruples in the ring \( \mathbb{Z}[\sqrt{-t}] \), with \( t>0 \). Their strong motivation for this research is that the extensibility of \( D(-1) \)-\( m \)-tuples is an active research topic from the vast recent literature.