Positive integer solutions of some second-order Diophantine equations.

Let k ≥ 3 be a positive integer and A = k ∓ 2. In this paper, we give all positive integer solutions of the second-order diophantine equations x² - kxy+y² = ∓ A, x² - (k² - 4)y² = ∓4A, x² - kxy+y² = ∓ (k² -4)A, x² - (k² +2)xy+y² = k² , x² (k² ∓2)xy+y² = k², and x² +4xy -[(k² 2)y]² = 4k² in terms of generalized Fibonacci and Lucas sequences. Moreover, we find necessary and sufficient condition for the equations x² - kxy + y² = -(k + 2) and x² - kxy + y² = k - 2 to have integer solutions. [ABSTRACT FROM AUTHOR]