Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-18T13:43:57.800Z Has data issue: false hasContentIssue false
  • English
  • Français

Pairs of additive equations III: quintic equations

Published online by Cambridge University Press:  20 January 2009

R. J. Cook
R. J. Cook
Affiliation:
University of Sheffield, Sheffield 10.
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.

We consider R simultaneous equations of additive type

where the coefficients aij are integers. Artin's conjecture, for additive forms, is that the equations (1) have a non-trivial solution in integers x1,…,xN provided that they have a non-trivial real solution, which is clearly satisfied when k is odd, and

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 26 , Issue 2 , June 1983 , pp. 191 - 211
Copyright
Copyright © Edinburgh Mathematical Society 1983

References

REFERENCES

1.Chowla, S. and Shimura, G., On the representation of zero by a linear combination of kth powers, Norske Vid. Selsk. Forh. 36 (1963), 169–178.Google Scholar
2.Cook, R. J., Simultaneous quadratic equations, J. London Math. Soc. 4 (1971), 319–326.CrossRefGoogle Scholar
3.Cook, R. J., A note on a lemma of Hua, Quart. J. Math. (Oxford) 23 (1972), 287–288.Google Scholar
4.Cook, R. J., Pairs of additive equations, Michigan Math. J. 19 (1972), 325–331.CrossRefGoogle Scholar
5.Cook, R. J., Pairs of additive equations II: large odd degree, J. Number Theory (to appear).Google Scholar
6.Cook, R. J., Simultaneous additive congruences, J. Number Theory (submitted).Google Scholar
7.Davenport, H., On sums of positive integral kth powers, Proc. Roy. Soc. 170A (1939), 293–299.Google Scholar
8.Davenport, H. and Lewis, D. J., Homogeneous additive equations, Proc. Roy. Soc. 274A (1963), 443–460.Google Scholar
9.Davenport, H. and Lewis, D. J., Cubic equations of additive type, Phil. Trans. Roy. Soc. 261A (1966), 97–136.Google Scholar
10.Davenport, H. and Lewis, D. J., Two additive equations, Proc. Sympos. Pure Math. XII(Amer. Math. Soc, 1969), 74–98.Google Scholar
11.Davenport, H. and Lewis, D. J., Simultaneous equations of additive type, Phil. Trans. Roy. Soc. 264A (1969), 557–595.Google Scholar
12.Tietäväinen, A., On a problem of Chowla and Shimura, J. Number Theory 3 (1971), 247–252.CrossRefGoogle Scholar
13.Vaughan, R. C., Homogeneous additive equations and Waring's problem, Acta Arith. 33(1977), 231–253.CrossRefGoogle Scholar
14.Vaughan, R. C., On pairs of additive cubic equations, Proc. London Math. Soc. 34 (1977), 354–364.CrossRefGoogle Scholar
15.Vaughan, R. C., The Hardy-Littlewood Method (Cambridge University Press, 1981).Google Scholar
16.Vinogradov, I. M., The Method of Trigonometrical Sums in the Theory of Numbers, translated and revised by Roth, K. F. and Davenport, A. (Interscience, London, 1954).Google Scholar
17.Vinogradov, I. M., On an upper bound for G(n), Izvestia Akad. Nauk. SSSR 23 (1959), 637–642.Google Scholar