A NOTE ON THE DIOPHANTINE EQUATION lx(3) - kx(2)

In this work, we investigate the Diophantine equation lx(3) - kx(2) + kx - l = y(2) where k and l are positive integers. The two results are Theorems 1.1 and 1.2. The first theorem states that if k = 3l - 1 and l = rho(2), the above equation has a unique integer solution, namely (x, y) = (1, 0). The second theorem says that if k = 3l + 1 and l equivalent to 0, 1, 4, 5, 7 (mod 8) the above equation also has a unique solution, the pair (x, y) = (1, 0).