Admissible Pairs: A Variation to Pollock Conjectures
We introduce a notion of an admissible pair, and prove a variation to Pollock's conjectures on icosahedral and dodecahedral numbers.
math.NT↗
arXiv subjects
Vi Anh Nguyen
Publications and source records attributed to Vi Anh Nguyen.
Erd\H{o}s-Moser Equation in Arithmetic Progressions
We consider the Erd\H{o}s-Moser equation $1^k+2^k+\cdots+(m-1)^k=m^k$ in arithmetic progressions. We prove among other things that when $k=2$, for any solution to exist, the above sum in arithmetic progression must consist of two or four terms. In either case, there are infinitely many solutions that can be completely characterized.
math.NT↗
Effective Estimates for a Class of Farey Fraction Sums and Bounds for Mundici-Type Constants
Let $D_{2}(Q)$ denote the sum of squared distances between consecutive Farey fractions in the full interval $(0, 1]$. Daniele Mundici conjectured that $C(Q):=D_{2}(Q)\cdot Q^2/\log Q$ is less than 3 for all $Q\geq 2$, which is confirmed true in \cite{DLN2026}. In this paper, we generalize this result to subintervals of $(0, 1]$ and to $h$-spacings. As applications, we obtain Mundici-type bounds in these two settings, extending the full-interval consecutive-spacing case of Mundici's conjecture.
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On a Conjecture about Sums Involving Farey Fractions
In this paper, we prove a conjecture by Daniele Mundici on the sum of squared distances between consecutive elements in the $Q$-th Farey sequence for $Q\in\mathbb{Z}$ and $Q\geq 2$.
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Characterization of Well-Totally Dominated Trees
Let $G$ be a graph with no isolated vertices. A set of vertices $S$ is a total dominating set (TDS) if every vertex in $G$ is adjacent to at least one vertex in $S$. We say $G$ is well-totally dominated (WTD) if every minimal TDS has the same size. In this paper, we present two characterizations of well-totally dominated trees, one being descriptive and the other being constructive. In particular, our characterizations imply that it takes only polynomial time to verify whether a given tree is WTD.
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