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The Number of Solutions of Congruences in Incomplete Residue Systems

Published online by Cambridge University Press:  20 November 2018

J. H. H. Chalk
J. H. H. Chalk*
Affiliation:
University of Toronto
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Let denote the set of points x = (x1, . . . , xn) with integral co-ordinates in Euclidean n-space. For any fixed integer m ≥ 2, let C = C(m) be the set of such points in the cube 0 ≤ xi < m (i = 1, 2, . . . , n) and let be any subset of C. Suppose that f(x) is any single-valued, integral-valued function, defined for all x ∊ . We consider solutions x ∊ of the congruence

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 15 , 1963 , pp. 291 - 296
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Carlitz, L. and Ucheyama, S., Bounds for exponential sums, Duke Math. J., 34 (1957), 37–41.Google Scholar
2. Hua, L. K. and Vandiver, H. S., Characters over certain types of rings with applications to the theory of equations in a finite field, Proc. Natl. Acad. Sci. U.S.A., 35 (1949), 94–99.Google Scholar
3. Mordell, L. J., The number of solutions of some congruences in two variables, Math. Zeitsch., 37 (1933), 193–209.Google Scholar
4. Mordell, L. J., Qn the number of solutions in incomplete residue sets of quadratic congruences, Archiv der Math., 8 (1957), 153–157.Google Scholar
5. Vinogradov, I. M., Elements of number theory (Dover, 1954).Google Scholar
6. Weil, A., On some exponential sums, Proc. Natl. Acad. Sci. U.S.A., 34 (1948), 205–207.Google Scholar
7. Weil, A., Number of solutions of equations in finite fields, Bull. Amer. Math. Soc, 55 (1949), 497–508.Google Scholar
8. Weil, A., On theRiemann hypothesis in function fields, Proc. Natl. Acad. Sci. U.S.A., 27 (1941), 345–347.Google Scholar