A NOTE ON SOME DIOPHANTINE EQUATIONS

Abstract

Let k >= 3 be an odd integer. In this paper we investigate all positive integer solutions of the equations x(4) - kx(2) y + y(2) = -/+ A, x(4) kx(2) y + y(2) = -/+ A(k(2) - 4), x(4) - (k(2) - 4)y(2) = -/+ 4A, and x(2) - (k(2) - 4)y(4) = -/+ 4A with A = -/+(k -/+ 2). We show that if k = 1(mod8) and k(2) - 4 be a square-free integer; then the equation x(4) - kx(2) y + y(2) = (k - 2)(k(2) - 4) has no positive integer solutions. Moreover, if k(2) - 4 be a square-free integer; then the equation x(4) - kx(2) y + y(2) = -(k + 2)(k(2) - 4) has no positive integer solutions.