A Pellian Equation with Primes and Applications to $$D(-1)$$ -Quadruples
Tóm tắt
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
$${\mathbb {Z}}[\sqrt{-t}], t>0$$
.
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