Solutions Of Negative Pell’s Equation Involving Interprimes

Abstract

Many researchers have been devoted to finding the solutions (𝜼,𝜻) in the set of non-negative integers, of Diophantine equation (Pell Equation) of the type 𝜼𝟐=𝑫𝜻𝟐±𝜶, where the value 𝜶 is fixed positive integers. In this article, we look for non-trivial integer solutions to two negative Pell equations 𝒙𝟐=𝟐𝟔𝒚𝟐−𝟗𝒕,and 𝒙𝟐=𝟓𝟎𝒚𝟐−𝟑𝟒𝒕, 𝒕∈𝑵 for the different choices of 𝒕 particular by (𝒊) 𝒕=𝟏, (𝒊𝒊) 𝒕=𝟑,(𝒊𝒊𝒊) 𝒕=𝟓,(𝒊𝒗) 𝒕=𝟐𝒌,(𝒗) 𝒕=𝟐𝒌+𝟓,∀ 𝒌∈𝑵. Additionally, recurrence relations on the solutions are obtained.