arXiv subjects

Ting Hon Stanford Li

Publications and source records attributed to Ting Hon Stanford Li.

4 recordsLinked to original sources

arXiv2026

The Density and Quantitative Upper Bound of Integers Not Expressible as the Sum of a Prime and a Zumkeller Number

A Zumkeller number is a positive integer whose set of positive divisors can be partitioned into two disjoint subsets of equal sums. In this paper, we investigate the additive properties of these numbers in conjunction with prime numbers. Specifically, we study the natural density of the exceptional set $\mathcal{E}$ consisting of all positive integers not expressible as the sum of a prime and a Zumkeller number. Let $α(x) = \#\{n \le x : n \in \mathcal{E}\}$ denote its counting function. We establish a quantitative upper bound for $α(x)$, demonstrating that $α(x) \ll x(\log \log x)^{-A}$, where $A = \log_2\left(\frac{2π^2}{2π^2-3}\right) \approx 0.2378$, thereby proving that the natural density of this exceptional set is $0$.

math.NT↗

arXiv2026

A Short Proof of a Conjecture Regarding Quadratic Representations of Practical Numbers

A positive integer $n$ is called a practical number if every positive integer less than or equal to $n$ can be expressed as a sum of distinct positive divisors of $n$. In this paper, we study an open conjecture proposed by Wang and Sun concerning the quadratic representations of practical numbers. Specifically, we provide a short proof of the second part of the conjecture, demonstrating that for any positive integers $b$ and $c$ with $2 \nmid b$ and $2 \mid c$, there exists an integer $n$ satisfying $1 < n \le \max\{b, c\}$ such that $n^2 + bn + c$ is a practical number. Combined with the work of Somu, Li, and Kukla, this completely settles the conjecture of Wang and Sun.

math.NT↗

arXiv2026

On a Generalization of Zumkeller Numbers: Infinitude of odd $k$-IGMO numbers for all non-negative integers $k$

In this paper, we introduce and investigate a novel generalization of Zumkeller numbers termed $k$-IGMO numbers, inspired by a problem proposed in the International Gamma Mathematical Olympiad (IGMO) 2025. A positive integer n is defined as a $k$-IGMO number if its set of positive divisors can be partitioned into two disjoint subsets whose elements have sums differing by $k$. Under this definition, classical Zumkeller numbers correspond to the case where $k = 0$. The main result of the paper is the existence of infinitely many odd $k$-IGMO numbers for all non-negative integers $k$.

math.GM↗

arXiv2022

On Some Results on Practical Numbers

A positive integer $n$ is said to be a practical number if every integer in $[1,n]$ can be represented as the sum of distinct divisors of $n$. In this article, we consider practical numbers of a given polynomial form. We give a necessary and sufficient condition on coefficients $a$ and $b$ for there to be infinitely many practical numbers of the form $an+b$. We also give a necessary and sufficient for a quadratic polynomial to contain infinitely many practical numbers, using which we solve first part of a conjecture mentioned in [9]. In the final section, we prove that every number of $8k+1$ form can be expressed as a sum of a practical number and a square, and for every $j\in \{0,\ldots,7\}\setminus \{1\}$ there are infinitely many natural numbers of $8k+j$ form which cannot be written as sum of a square and a practical number.

math.NT↗