Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T12:57:03.206Z Has data issue: false hasContentIssue false

On a Theorem of Aubry-Thue

Published online by Cambridge University Press:  20 November 2018

Alfred Brauer
Affiliation:
University of North Carolina
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.

In 1913 L. Aubry [1] proved the following theorem:

If a and m are relatively prime, m > 0, and if is not an integer, then it is always possible to find integers x and y not both zero such that

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1951

References

1. Aubry, L., Un théoréme d'arithmétique, Mathesis (4), vol. 3 (1913).Google Scholar
2. Ballieu, R., Sur des congruences arithmétiques, Bulletin de la Classe des Sciences de l'Académie Royale de Belgique (5), vol. 34 (1948), 3945.Google Scholar
3. De Backer, S. M., Un théorémefundamental, Bulletin de la Classe des Sciences de l'Académie Royale de Belgique (5), vol. 33 (1947) 632634.Google Scholar
4. De Backer, S. M., Solutions modérées déun systéme de congruences du premier degré pour un module premier p. Bulletin de la Classe des Sciences de l'Académie Royale de Belgique, (5) vol. 34 (1948), 4651.Google Scholar
5. Ore, O., Number theory and its history (New York, 1948), 268.Google Scholar
6. Porcelli, P. and Pall, G., A property of Farey sequences, Can. J. Math., vol. 3 (1951) 5253.Google Scholar
7. Scholz, A., Einfuhrung in die Zahlentheorie (Berlin, 1939).Google Scholar
8. Sylvester, J. J., Note on a principle in the theory of numbers and the resolubility of any number into the sum of four squares, Quar. J. of Math., vol. 1 (1857), 196-7; or Collected Math. Papers, vol. 2 (1908), 101102.Google Scholar
9. Thue, A., Über die ganzzahlige Gleichung , Norske videnskaps-akademi, Oslo, Matematisk-naturvidenskapelig klasse Skrifter, No. 3 (1915).Google Scholar
10. Thue, A. Et bevis for at lignigen A3 + B3= C3 er remulig i helefra nul forsk jellige tal A, B, og B.Archiv. for Math, og Naturvid, vol. 34, No. 15 (1917).Google Scholar
11. Uspensky, J. V. and Heaslit, M. A., Elementary number theory (New York, 1939).Google Scholar
12. Vinogradov, J. M., On a general theorem concerning the distribution of the residues and non-residues of powers, Trans. Amer. Math. Soc, vol. 29 (1927), 209–17.Google Scholar
13. Vinogradov, J. M. On the bound of the least non-residues of nth powers, Trans. Amer. Math. Soc. vol. 29 (1927), 218226.Google Scholar