ON CONSECUTIVE INTEGERS OF THE FORM ax2; by2 AND cz2

have a solution in positive integers (x, y, z) other than that given by x = 5, y = 7 and z = 4. A negative answer to this question follows from a classical result of Ljunggren [8], as recently refined by Cohn [4]: Theorem 1.1. Let the fundamental solution of the equation v2−Du2 = 1 be a + b √ D (i.e. (v, u) = (a, b) is the smallest positive solution). Then the only possible solutions of the equation x4−Dy2 = 1 are given by x2 = a and x2 = 2a2 − 1; both solutions occur in only one case, D = 1785. To see this, note that (1) implies that y4 − 6(xz)2 = 1. More generally, if a, b and c are positive integers, one may consider the simultaneous Diophantine equations