Date of Award

Spring 5-15-2015

Degree Type

Thesis

Degree Name

M.S.

Degree Program

Mathematics

Department

Mathematics

Major Professor

Craig A. Jensen

Second Advisor

Kenneth Holladay

Third Advisor

Ralph Saxton

Abstract

Motivated by primality and integer factorization, this thesis introduces generalizations of standard binary multiplication to commutative n-ary operations based upon geometric construction and representation. This class of operations are constructed to preserve commutativity and identity so that binary multiplication is included as a special case, in order to preserve relationships with ordinary multiplicative number theory. This leads to a study of their expression in terms of elementary symmetric polynomials, and connections are made to results from the theory of polyadic (n-ary) groups. Higher order operations yield wider factorization and representation possibilities which correspond to reductions in the set of primes as well as tiered notions of primality. This comes at the expense of familiar algebraic properties such as associativity, and unique factorization. Criteria for primality and a naive testing algorithm are given for the ternary arithmetic, drawing heavily upon modular arithmetic. Finally, connections with the theory of partitions of integers and quadratic forms are discussed in relation to questions about cardinality of primes.

Rights

The University of New Orleans and its agents retain the non-exclusive license to archive and make accessible this dissertation or thesis in whole or in part in all forms of media, now or hereafter known. The author retains all other ownership rights to the copyright of the thesis or dissertation.

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Creative Commons Attribution-Noncommercial 4.0 License
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