On the Diophantine equation $x^2 - kxy + y^2 + 2^n = 0$

Abstract

In this paper, we determine when the equation in the title has an infinite number of positive integer solutions x and y when 0 Ä n Ä 10: Moreover, we give all the positive integer solutions of the same equation for 0 Ä n Ä 10:

Proof of the conjecture of Keskin, Siar and Karaatli

“…(ii) Suppose that n is even and that the equation x 2 − kxy + y 2 = −2 n has a positive integer solution, then clearly k is even because n ≥ 1, (the case n = 0 has been settled in [2]). Again the same method in the proof of Theorem 3.1 and Lemma 2.2 imply that k ≤ 2 n + 2 and k even.…”

“…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. Yuan and Hu [6] considred equation (1), with l = 2 or 4 and determined the values of the integer k for which equation (1) has an infinite number of positive solutions.…”

“…Yuan and Hu [6] considred equation (1), with l = 2 or 4 and determined the values of the integer k for which equation (1) has an infinite number of positive solutions. Expanding on the work of Yuan and Hu [6], Keskin et al in [2] and [3] considered equation (1) for l = ±2 r , with r a positive integer. They explained that in order to determine when equation (1) with l = −2 r , has an infinite number of positive integer solutions, one needs only to determine when the diophantine equation…”

“…Keskin et al solved equation (2) and equation (3) for 0 ≤ n ≤ 10, and formulated the following conjecture.…”

“…(ii) Let n be an even integer. If k > 2 n −2, then equation (2) has no positive odd integer solution. If k ≤ 2 n −2 and equation (2) has a positive odd integer solution, then k is even.…”