arXiv subjects
Publications and source records attributed to Jeff Rushall.
The use of quadratic residues to construct matrices with specific determinant values is a familiar problem with connections to many areas of mathematics and statistics. Our research has focused on using cubic residues to construct matrices with interesting and predictable determinants.
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.
A simple and connected $n$-vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers $1, 2, 3,\ldots, n$, such that adjacent vertices have relatively prime labels. We will present previously unknown prime vertex labelings for new families of graphs including cycle pendant stars, cycle chains, prisms, and generalized books.
This paper considers numerical semigroups $S$ that have a non-principal relative ideal $I$ such that $μ_S(I)μ_S(S-I)=μ_S(I+(S-I)) $. We show the existence of an infinite family of such which $I+(S-I)=S\backslash\{0\}$. We also show examples of such pairs that are not members of this family. We discuss the computational process used to find these examples and present some open questions pertaining to them.