Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T23:58:25.265Z Has data issue: false hasContentIssue false
  • English
  • Français

Small Solutions of the Congruence ax2 + by2 ≡ c(mod k)

Published online by Cambridge University Press:  20 November 2018

Kenneth S. Williams
Kenneth S. Williams*
Affiliation:
Carleton University
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 1957, Mordell [3] proved

Theorem. If p is an odd prime there exist non-negative integers x, y ≤ A p3/4 log p, where A is a positive absolute constant, such that

(1.1)

provided (abc, p) = 1.

Recently Smith [5] has obtained a sharp asymptotic formula for the sum where r(n) denotes the number of representations of n as the sum of two squares.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 3 , June 1969 , pp. 311 - 320
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Estermann, T., On Kloosterman's Sum. Mathematika 8 (1961) 83–86.Google Scholar
2. Estermann, T., A new application of the Hardy-Littlewood-Kloostermann Method. Proc. Lond. Math. Soc. 12 (1962) 425–444.Google Scholar
3. Mordell, L. J., On the number of solutions in incomplete residue sets of quadratic congruences. Archiv der Math. 8 (1957) 153–157.Google Scholar
4. Rademacher, H., Lectures on Elementary Number Theory. (Blaisdell, 1964) 93.Google Scholar
5. Smith, R. A., The circle problem in an arithmetic progression. Canad. Math. Bull. 11 (1968) 175–184.Google Scholar
6. Tietäväinen, A., On the trace of a polynomial over a finite field. Ann. Univ. Turku., Ser. Al, 87 (1966) 3–7.Google Scholar
7. Tietäväinen, A., On non-residues of a polynomial. Ann. Univ. Turku., Ser. Al, 94 (1966) 3–6.Google Scholar
8. Weil, A., On some exponential sums. Proc. Nat. Acad. Sci. (U.S.A.) 34 (1948) 204–207.Google Scholar