The distribution of solutions

of the congruence

Author:
Anwar Ayyad

Journal:
Proc. Amer. Math. Soc. **127** (1999), 943-950

MSC (1991):
Primary 11D79, 11L40

DOI:
https://doi.org/10.1090/S0002-9939-99-05124-2

MathSciNet review:
1641700

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Abstract | References | Similar Articles | Additional Information

Abstract: For a cube of size , we obtain a lower bound on so that is nonempty, where is the algebraic subset of defined by

a positive integer and an integer not divisible by . For we obtain that is nonempty if , for we obtain that is nonempty if , and for we obtain that is nonempty if . Using the assumption of the Grand Riemann Hypothesis we obtain is nonempty if .

**[1]**A. Ayyad, T. Cochrane, and Z. Zheng,*The congruence , the equation , and mean values of character sums*, J. of Number Theory**59**(2) (1996), 398-413. MR**97i:11091****[2]**D.A. Burgess,*On character sums and primitive roots*, Proc. London Math. Soc.(3)**12**(1962), 179-192. MR**24:A2569****[3]**H.L. Montgomery and R.C. Vaughan,*Exponential sums with multiplicative coefficients*, Invent. Math.**43**(1977), 69-82. MR**56:15579****[4]**R.A. Smith,*The distribution of rational points on a curve modulo*, Rocky Mountain J. of Math.**15**(2) (1985), 589-597. MR**87h:11055**

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Additional Information

**Anwar Ayyad**

Affiliation:
Department of Mathematics, Kansas State University, Manhattan, Kansas 66506

Address at time of publication:
Department of Mathematics, University of Gaza, P.O. Box 1418, Gaza Strip, Via Israel

Email:
anwar@math.ksu.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-05124-2

Keywords:
Distribution,
congruences,
solutions

Received by editor(s):
May 9, 1997

Communicated by:
Dennis A. Hejhal

Article copyright:
© Copyright 1999
American Mathematical Society