On consecutive integer solutions for $y^{2}-k=x^{3}$

Abstract

In this paper the number of k's having a given number of consecutive integer solutions either for x or for y or for both in the 2 3 equation y k = x has been found. Introduction. The equation y2 k =x3 now known as Mordell's equation, has interested mathematicians for more than three centuries, and has played an important role in the development of number theory. Many interesting, important, and beautiful results on this subject have been published. For a complete bibliography one can see [1]. It is really surprising to see that an interesting problem like finding the number of k's having a given number of solutions of a given type has been overlooked by all for such a long time. In a recent paper [2] we have given the identity ( t33) 223 (t3 + 1)2 (2t)3 = (3t3 _ 1)2 (2t2)3 = t6 6t3 + 1. For even t this shows that there is an infinite number of k's having six solutions (x, ?y) where x and y are coprime. This result also follows from the identity (4a 3 + 1)2(2a)3 = (4a3)2(-1)3 = (4a3 1)2 (2a3 = 16a6 + 1, which is obtained in this paper. We want to find the number of k's having a given number of consecutive integer solutions for y or x or both. Theorem 1. If y2 _ k = x3 (k an integer) has two consecutive integer solutions for y, then k is a positive integer. Proof. If x is a negative integer, then k = y2 _ x3 is always positive. So we may assume that x takes only positive integer values. Let (x1, y1) and (x2, y1 -1) be two integer solutions for y2 k = x3. Then we have Received by the editors November 5, 1973. AMS (MOS) subject classifications (1970). Primary lOB1O. 'This paper is dedicated to Professor M. T. Nagell, Uppsala University, Sweden. Copyright ? 1975, American Mathematical Society