Let $G(m,k)=\max\lbrace \text{GCD}((n+1)^m+k,n^m+k)\ |\ n \in \mathbb N \rbrace$. The fact that $G(1,k)=1$ is trivial and Berzsenyi has shown that $G(2,k)=|4k+1|$. We give explicit formulas for $G(m,k)$ for $m=3,4,5$.

The pure symmetric automorphism group, $\text{P}\Sigma _n$, consists of those automorphisms of the free group on $n$ generators that take each standard generator to a conjugate of itself. We give presentations for kernels of homomorphisms $\text{P}\Sigma _n\to \mathbb{Z}$ where each standard generator is sent to either 0 or 1, and provide explicit generators (as words in the standard generators) when those kernels are finitely generated. In addition, we provide recursive constructions of the defining graphs of the graph groups associated with $\text{P}\Sigma _n$.

We make a comparison between the identity matrix and the counteridentity matrix by studying the eigensystem of $J+M$, where $J$ is the counteridentity matrix and $M$ is a structured matrix, with the goal of expressing this system entirely in terms of the corresponding eigensystem of $M$. Also, properties of some parametric families of the form $J+\rho M$ are studied.

In this work, we define a spherical taxicab distance on the surface of a sphere in terms of well-known geographical latitude and longitude. Then we study some properties of this distance function and give an analogue of the ruler postulate. At the end, we give some connections between the spherical taxicab circle and the spherical circles.