On the Diophantine equation x 2 − kxy + y 2 − 2 n = 0

Abstract

Abstract. In this study, we determine when the Diophantine equation x 2 −kxy+y 2 −2 n = 0 has an infinite number of positive integer solutions x and y for 0 n 10. Moreover, we give all positive integer solutions of the same equation for 0 n 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x 2 − kxy + y 2 − 2 n = 0.

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.…”

“…Remark 3.1. It was proved in [3], that the diophantine equation x 2 −kxy + y 2 = 2 n with k = 2 n − 2 has infinitly many solutions and in [2], that the Diophantine equation x 2 − kxy + y 2 = −2 n , with k = 2 n + 2 has infinitly many solutions. Hence the bounds of k in Theorem 3.1 and Theorem 3.2 are sharp.…”

“…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…”

On the Exponential Diophantine Equation $(a^2-2)(b^2-2)=x^2$

“…where a > 1 and b > 1 are different fixed integers. Firstly, Szalay [13] handled this equation for (a, b) = (2, 3), (2,5), and (2, 2 k ). He showed that there is no solution if (a, b) = (2, 3), only the solution n = 1 for (a, b) = (2, 5), and only the solution k = 2 and n = 3 for (a, b) = (2, 2 k ) with k > 1.…”