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A diophantine equation

Published online by Cambridge University Press:  18 May 2009

J. W. S. Cassels
J. W. S. Cassels
Affiliation:
D.P.M.M.S., University of Cambridge, 16 Mill Lane Cambridge CB2 1SB
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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

y2=(x−1)3+x3+(x+1)3

=3x(x2+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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 27 , October 1985 , pp. 11 - 18
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

REFERENCES

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