Hostname: page-component-89b8bd64d-z2ts4 Total loading time: 0 Render date: 2026-05-08T06:32:22.691Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Information

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

References

REFERENCES

1.Baker, A., Transcendental number theory (Cambridge, 1975).CrossRefGoogle Scholar
2.Baker, A. and Coates, J., Integer points on curves of genus 1, Proc. Cambridge Philos. Soc. 67 (1970), 595–602.CrossRefGoogle Scholar
3.Baker, A. and Davenport, H., The equations 3x 2−2=y 2 and 8x 2−7=z 2, Quart. J. Math. Oxford Ser. 2, 20 (1969), 129–37.CrossRefGoogle Scholar
4.Cassels, J. W. S., Integral points on certain elliptic curves, Proc. London Math. Soc. (3) 14A (1965), 55–57.CrossRefGoogle Scholar
5.Ellison, F., Ellison, W. J., Pesek, J.Stall, C. E. and Stall, D. S., The diophantine equation y 2+k=x 3, J. Number Theory 4 (1972), 107–117.CrossRefGoogle Scholar
6.Hoare, G., Solution and comments on 67.A and 67.B, Math. Gaz. 67 (1983), 228–230.Google Scholar
7.Ljunggren, W., Zur Theorie der Gleichung x 2+1=Dy 4, Avh. Norske Vid.-Akad. Oslo, 1942 No. 5, 1.Google Scholar
8.Mordell, L. J., Diophantine equations (Academic Press, 1969).Google Scholar
9.Skolem, T., The use of p-adic methods in the theory of diophantine equations, Bull. Soc. Math. Belg. 7 (1955), 83–95.Google Scholar

An addendum has been issued for this article:

Addendum