Publication Date

5-5-1972

Abstract

In this paper we consider three unrelated problems in combinational number theory. The first of these pertains to minimal residual polynomials, the second to certain group factorizations and the third to the addition of positive integers in their Zeckendorf representations.

In section II We show that for any given positive integer m there exists a unique integer monic polynomial P of least degree and whose coefficients satisfy certain inequalities with the property that m divides P(x) for every integer x. This polynomial is called the minimal residual polynomial modulo m. We then show that this polynomial assumes various forms which depend upon the prime divisors of m.

A factorization of a groupoid (S,’) is a pair of nonempty subsets A and B of S with the property that each S e S can be uniquely represented as s=a*b where a e A and b e B. In Section III we first settle a conjecture of A. M. Vaidya by showing that if (A,B) is a factorization of a finite abelian group then |A n B|=1. We then successively determine |A n B| for factorizations of the additive groups of integers, rational numbers and real numbers.

Zeckendorf has shown that every positive integer m can be uniquely represented as a sum of distinct non-consecutive Fibonacci numbers. We call such a sum the Zeckendorf representation of m. The final problem in this paper, which is considered in Section IV, is the arithmetic of positive integers in Zeckendorf representation. Given any two positive integers a and b in their Zeckendorf representations we constructed an algorithm which yields the Zeckendorf representation of a+b. This type of addition, which will be shown to enjoy many of the properties of the ordinary addition of positive integers, has the unusual property that in general we have both “left” and “right carries”. Utilizing the methods of this algorithm we then characterize some of the distinctive features of this arithmetic.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Roger Charles Entringer

Second Committee Member

Gustave A. Efroymson

Third Committee Member

Theodore N. Guinn

Fourth Committee Member

Alexander Paul Stone

Language

English

Document Type

Dissertation

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