ON THE DIOPHANTINE EQUATIONS D1X(2)+2(2M)D2=Y(N) AND D1X(2)+D2=4Y(N)

Let d1, d2 be coprime positive integers, which are squarefree, and let h denote the class number of the imaginary quadratic field Q(square-root -d1d2). Let m, n be integers such that m greater-than-or-equal-to 0, n > 1, and gcd(n, 2h) = 1 . In this paper we prove that if n greater-than-or-equal-to 8.5 . 10(6), then the equations d1x2 + 2(2m)d2 = y(n) (2 dagger y) and d1x2 + d2 = 4y(n) have no positive integer solutions (x, y) with gcd(x, y) = 1.