Hostname: page-component-54dcc4c588-br6xx Total loading time: 0 Render date: 2025-09-11T14:32:30.665Z Has data issue: false hasContentIssue false
  • English
  • Français

A theorem of Minkowski on the product of two linear forms

Published online by Cambridge University Press:  24 October 2008

J. H. H. Chalk
J. H. H. Chalk
Affiliation:
Bedford CollegeLondon, N.W.1
Get access Rights & Permissions [Opens in a new window]

Extract

1. Let Λ denote any plane lattice of determinant Δ ╪ 0. Then it is well known that the region

contains a pair of generating points of Λ; and that, unless the lattice Λ is of the special form

where both α│β, γ│δ are rational, there are an infinity of such pairs in the interior of the region. Minkowski (4) obtained this result by considering the ‘tangent’ parallelogram Пλ,

inscribed in the region (1). With simple geometrical arguments, he showed that

(i) Пλ always contains a point of Λ, other than 0,

(ii) if Пλ contains two primitive points of Λ then these generate Λ, except possibly when Λ has points, other than 0, on both coordinate axes.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 49 , Issue 3 , July 1953 , pp. 413 - 420
Copyright
Copyright © Cambridge Philosophical Society 1953

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

(1)Dickson, L. E.Introduction to the theory of numbers (Chicago, 1929).Google Scholar
(2)Koksma, J. F.Diophantische Approximationen (Berlin, 1936), chap. 3, §4.Google Scholar
(3)Lagrange, J. L.Oeuvres (Paris, 1869), vol. 3, pp. 693–758.Google Scholar
(4)Minkowski, H.Math. Ann. 54 (1900), 91–124.CrossRefGoogle Scholar
(5)Mordell, L. J.J. Lond. math. Soc. 3 (1928), 19–22.CrossRefGoogle Scholar
(6)Perron, O.Math. Ann. 115 (1938), 656–7.CrossRefGoogle Scholar