Number Theory Commons^™

Factorization Lengths In Numerical Monoids, Maya Samantha Schwartz

Senior Projects Spring 2019

A numerical monoid M generated by the natural numbers n_1, ..., n_k is a subset of {0, 1, 2, ...} whose elements are non-negative linear combinations of the generators n_1, ..., n_k. The set of factorizations of an element in M is the set of all the different ways to write that element as a linear combination of the generators. The length of a factorization of an element is the sum of the coefficients of that factorization. Since an element in a monoid can be written in different ways in terms of the generators, its set of factorization lengths may …

Commutative N-Ary Arithmetic, Aram Bingham

University of New Orleans Theses and Dissertations

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 …

On The Irreducibility Of The Cauchy-Mirimanoff Polynomials, Brian C. Irick

Doctoral Dissertations

The Cauchy-Mirimanoff Polynomials are a class of polynomials that naturally arise in various classical studies of Fermat's Last Theorem. Originally conjectured to be irreducible over 100 years ago, the irreducibility of the Cauchy-Mirimanoff polynomials is still an open conjecture.

This dissertation takes a new approach to the study of the Cauchy-Mirimanoff Polynomials. The reciprocal transform of a self-reciprocal polynomial is defined, and the reciprocal transforms of the Cauchy-Mirimanoff Polynomials are found and studied. Particular attention is given to the Cauchy-Mirimanoff Polynomials with index three times a power of a prime, and it is shown that the Cauchy-Mirimanoff Polynomials of index …