A diophantine equation

I was recently challenged to find all the cases when the sum of three consecutive integral cubes is a square; that is to find all integral solutions x, y of y 2 =( x −1) 3 + x 3 +( x +1) 3 =3 x ( x 2 +2) This is an example of a curve of genus 1. There is an effective procedure for finding all integral points on a given curve of genus 1 (1, Theorem 4.2, 2): that is, it can be guaranteed to find all the integral points and to show that no others exist with a finite amount of work. Unlike some effective procedures, which have only logical interest, this one can actually be carried out in practice, at least with the aid of a computer (3, 5). There are, however, older methods for dealing with problems of this kind which, while not effective, very often lead more easily to a complete set of solutions (and a proof that it is complete). I solve the problem here by a technique introduced in 4. It requires only the elementary theory of algebraic number–fields. The motivation is p –adic, but it is simpler not to introduce p –adic theory overtly.