engleski

A Pellian equation with primes and its applications

This is a joint work with Andrej Dujella and Mirela Juki\'c Bokun. Let $p$ be an odd prime and $k$ non-negative integer. We consider a Pellian equation of the form x^2-(p^{; ; 2k+2}; ; +1)y^2=-p^{; ; 2l+1}; ; , l \in \{; ; 0, 1, \dots, k\}; ; , and prove that it has no solutions in positive integers $x$ and $y$. By using this result and other known results on the topic of Diophantine $m$-tuples, we obtain results on extensibility of $D(-1)$-pairs of the form $\{; ; 1, 2b\}; ; $, where $2b= p^{; ; 2j}; ; +~1$, $j>0$, and $p$ is an odd prime, to $D(-1)$-quadruples in the ring $\mathbb{; ; Z}; ; [\sqrt{; ; -t}; ; ], t>0$.

Pellian equation ; Diophantine $m$-tuples ; Ring of integers in quadratic field

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