Graduate Research Theses & Dissertations

1966

Document Type

Dissertation/Thesis

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.

Includes bibliographical references.

29 pages

eng

Publisher

Northern Illinois University