# A Pellian equation with primes and applications to D(-1)-quadruples

## TL;DR

This paper proves a specific Pell-type equation with prime parameters has no positive integer solutions and uses this to study the extension of certain Diophantine pairs to quadruples in quadratic integer rings.

## Contribution

It establishes the non-solvability of a class of Pellian equations with prime parameters and applies this to analyze the extension of D(-1)-pairs to quadruples in quadratic integer rings.

## Key findings

- The Pell-type equation has no positive integer solutions.
- Certain D(-1)-pairs cannot be extended to quadruples in specified rings.
- Results contribute to understanding the structure of Diophantine quadruples.

## Abstract

In this paper, we prove that the equation $x^2-(p^{2k+2}+1)y^2=-p^{2l+1}$, $l \in \{0,1,\dots,k\}, k \geq 0$, 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 $\mathbb{Z}[\sqrt{-t}], t>0$.