Number theory is an area of mathematics not unknown to students majoring in mathematics teaching. As early as in junior high, they may encounter basic number theory concepts such as prime numbers, multiples, divisors, or even the fundamental theorem of arithmetic—the proof of which, however, will only be included in the university curriculum of mathematics teacher students. When teaching number theory, in addition to inspiring and motivating students, it is important to have them ask the right questions. Proofs are important, but so is the proper formulation of mathematical conjectures, not to mention intuitions regarding mathematical properties and relationships. These processes are facilitated by the software used, developed to visualize functions related to number theory. Our study aims to assess the relevance of visualization in the teaching and understanding of certain arithmetic functions among mathematics teacher students. Furthermore, we aim to investigate the practical usability of our visualization software, relying on student feedback. According to our results, students found the software useful, as it helped them to discover previously unknown relationships in the examined arithmetic functions. We may conclude our software fills a niche, with student feedback indicating its usefulness.

The aim of this article is to investigate the relations between the exponent of the convergence of sequences and other characteristics defined for monotone sequences of positive numbers. Another main goal is to characterize such monotone sequences $\left(a_n\right)$ of positive numbers that, for each $n\ge 2$, satisfy the equality $a_n=K(a_{n-1},a_{n+1})$, where the function $K:{\mathbb{R}}^{+}\times {\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is the mean, i.e., each value of $K(x,y)$ lies between $min\{x,y\}$ and $max\{x,y\}$. Well-known examples of such sequences are, for example, arithmetic (geometric) progression, because starting from the second term, each of its terms is equal to the arithmetic (geometric) mean of its neighboring terms. Furthermore, this accomplishment generalized and extended previous results, where the properties of the logarithmic sequence $\left(a_n\right)$ are referred to, i.e., in such a sequence that every $n\ge 2$ satisfies $a_n=L(a_{n-1},a_{n+1})$, where $L(x,y)$ is the logarithmic mean of positive numbers $x,y$ defined as follows: $L(x,y):=\left\{\begin{array}{ccc}{\displaystyle \frac{y-x}{lny-lnx}}\hfill & \mathrm{if}\hfill & x\ne y,\hfill \\ x\hfill & \mathrm{if}\hfill & x=y.\hfill \end{array}\right.$

Zero has (a) special role(s) in mathematics. In the current century, we take negative numbers and zero for granted, but we should also be aware that their acceptance and their emergence in mathematics, and their ubiquity today, have not come to happen as rapidly as, for example, that of natural numbers. Students can quickly become confused by the question: is zero a natural number? The answer is simple: a matter of definition. The history of zero and that of negative numbers are closely linked. It was in the calculations of debts that the negative numbers first appeared, where the state of balance between positive and negative assets was also given. We now take them for granted, but the history of science shows that it was far from a smooth process, and interesting mistakes about zero still occur in education today. I present a few examples below, and how they have been resolved.