Infinitely many positive solutions of the diophantine equation x2 − kxy + y2 + x = 0

On negative Pell equations: Solvability and unsolvability in integers

“…This work is devoted to constructing a class of positive integers d ≡ d(u n , u n+1 , m) expressed by u n , u n+1 and fourth-degree polynomials of m such that the negative Pell equation x 2 − dy 2 = −1 is solvable in integers, where {u n } n∈N∪{0} satisfies u 0 = u 1 = 1 and u n+2 = 3u n+1 − u n for n ∈ N ∪ {0}. On the other hand, due to related works [4,5], we are also interested in a special equation x 2 −k(k +4)m 2 y 2 = −1, where k, m ∈ N. A sufficient and necessary condition for the solvability of this equation in integers has been established. Before introducing the main ideas and the results, we recall the history of the Pell equation and some related works.…”

“…As an application of Theorem 1.1, we refer the reader to Examples 1-3 in Section 3. In [4,5], the authors ever studied the solvability in integers x and y for the following Pell equation x 2 − k(k + 4)y 2 = −4 by using an infinite simple continued fraction of k(k + 4), where k ∈ N. As a special case of (1.4) with a = km and b = (k + 4)m, we use a different approach to establish a sufficient and necessary condition for the solvability of…”

On the Diophantine Equation x2 – kxy + ky2 + ly = 0, l = 2n

“…They determined the value of k in the case of equation (1) has an infinite number of positive integer solutions. They found that equation (1) for l = 1 has infinitely many integer solutions if and only if k = 0 and ±1 which generalized the work of Marlewski and Zarzycki [4]. After that Hu and Le [1] considered equation (1) for non zero integer l. They characterized the value of positive integer k that makes equation (1) has infinitely many positive integer solutions.…”

Proof of the conjecture of Keskin, Siar and Karaatli

“…x 2 − kxy + y 2 + lx = 0 (1) for different values of the integers k and l. Marlewski and Zarzycki [4], considered equation (1) for l = 1, and proved that equation (1) has no positive solution for l = 1 and k > 3, but has an infinite number of solutions for k = 3 and l = 1. Keskin et al in [2] and [3] considered equation (1) for l = −1 and proved that it has positive integer solutions for k > 1.…”