#### Publication Date

1966

#### Document Type

Dissertation/Thesis

#### First Advisor

McKenzie, Harvey C.||Beach, James W.

#### Degree Name

M.S. (Master of Science)

#### Legacy Department

Department of Mathematics

#### LCSH

Number theory; Congruences and residues

#### Abstract

In this paper a study is first made of the congruence xⁿ ≡ b mod p, where p is a prime number, and particularly when n divides p - 1. Then some properties of primitive roots of prime numbers are investigated. In one chapter it is shown that xⁿ ≡ b mod p always has a solution if (n,p - 1) = 1. If n divides p - 1 there are (p-1)/n of these congruences that have solutions and these elements form a group. A method is given for determining whether or not the congruence has a solution if n divides p - 1. A theorem is proved which gives the number of solutions each of these congruences has, and then results are proved which give the sums of solution sets, the sums of sets of elements that have solutions and the sums of elements of certain groups. In another chapter, it is shown that the sum of the primitive roots of a prime is -1, 0, or 1 depending on certain conditions. Then, theorems are proved which allow us to find primitive roots of special classes of primes.

#### Recommended Citation

Corzatt, Clifton E., "On nth residues and primitive roots modulo a prime" (1966). *Graduate Research Theses & Dissertations*. 4180.

https://huskiecommons.lib.niu.edu/allgraduate-thesesdissertations/4180

#### Extent

29 pages

#### Language

eng

#### Publisher

Northern Illinois University

#### Rights Statement

In Copyright

#### Rights Statement 2

NIU theses are protected by copyright. They may be viewed from Huskie Commons for any purpose, but reproduction or distribution in any format is prohibited without the written permission of the authors.

#### Media Type

Text

## Comments

Includes bibliographical references.