A Pellian Equation with Primes and Applications to $$D(-1)$$ -Quadruples

In this paper, we prove that the equation $$x^2-(p^{2k+2}+1)y^2=-p^{2l+1}$$ , $$l \in \{0,1,\dots ,k\}, k \ge 0$$ , where p is an odd prime number, is not solvable in positive integers x and y. By combining that result with other known results on the existence of Diophantine quadruples, we are able to prove results on the extensibility of some $$D(-1)$$ -pairs to quadruples in the ring $${\mathbb {Z}}[\sqrt{-t}], t>0$$ .