arXiv subjects

Zachary Parker

Publications and source records attributed to Zachary Parker.

3 recordsLinked to original sources

arXiv2026

Ternary Perfect Forms Over Imaginary Quadratic Field

In this work, we enumerate the ternary perfect forms over all imaginary quadratic fields of absolute discriminant up to $91$ and study the number and combinatorial types of their associated polytopes. We prove a bound on the number of combinatorial types that can arise regardless of the discriminant. We analyze the data and make several observations concerning the number and complexity of the perfect forms that arise in the scope of our calculation.

math.NT↗

arXiv2020

A series of series topologies on $\mathbb{N}$

Each series $\sum_{n=1}^\infty a_n$ of real positive terms gives rise to a topology on $\mathbb{N} = \{1,2,3,...\}$ by declaring a proper subset $A\subseteq \mathbb{N}$ to be closed if $\sum_{n\in A} a_n < \infty$. We explore the relationship between analytic properties of the series and topological properties on $\mathbb{N}$. In particular, we show that, up to homeomorphism, $|\mathbb{R}|$-many topologies are generated. We also find an uncountable family of examples $\{\mathbb{N}_α\}_{α\in [0,1]}$ with the property that for any $α< β$, there is a continuous bijection $\mathbb{N}_β\rightarrow \mathbb{N}_α$, but the only continuous functions $\mathbb{N}_α\rightarrow \mathbb{N}_β$ are constant.

math.GN↗

arXiv2016

Perfect Numbers in the Ring of Eisenstein Integers

One of the many number theoretic topics investigated by the ancient Greeks was perfect numbers, which are positive integers equal to the sum of their proper positive integral divisors. Mathematicians from Euclid to Euler investigated these mysterious numbers. We present results on perfect numbers in the ring of Eisenstein integers.

math.NT↗