arXiv subjects

Shi-Chao Chen

Publications and source records attributed to Shi-Chao Chen.

6 recordsLinked to original sources

arXiv2025

On the 3-rank of the class group of quadratic fields

Let $n\ge1$, $r\ge0$ and $s\ge0$ be integers satisfying $4+r+3 s\le3^{n+1}$. Given linear polynomials $f_{i}(x)=m_{i} x+n_{i}$ for $1 \le i \le r+s$, where the coefficients $m_{i} , n_{i}$ are positive integers satisfying certain conditions, we prove that there exist infinitely many fundamental discriminants $D>0$ such that the 3-rank of the class group of each quadratic fields $\mathbb{Q}(\sqrt{f_1(D)}), \ldots, \mathbb{Q}(\sqrt{f_r(D)})$ and $\mathbb{Q}(\sqrt{-f_{r+1}(D)}), \ldots, \mathbb{Q}(\sqrt{-f_{r+s}(D)})$ is simultaneously less than $n$. Moreover, for any positive integer $k$, there exist positive integers $a, d$ such that the 3-rank of the class group of each quadratic fields $\mathbb{Q}(\sqrt{a+g_1(d)}), \ldots,\mathbb{Q}(\sqrt{a+g_k(d)})$ is simultaneously less than $n$ for polynomials $g_1(x), g_2(x), \ldots, g_k(x)$ that take integer values at the integers and have no constant terms.

math.NT↗

arXiv2023

Partition theorems and the Chinese remainder theorem

The famous partition theorem of Euler states that partitions of $n$ into distinct parts are equinumerous with partitions of $n$ into odd parts. Another famous partition theorem due to MacMahon states that the number of partitions of $n$ with all parts repeated at least once equals the number of partitions of $n$ where all parts must be even or congruent to $3 \pmod 6$. These partition theorems were further extended by Glaisher, Andrews, Subbarao, Nyirenda and Mugwangwavari. In this paper, we utilize the Chinese remainder theorem to prove a comprehensive partition theorem that encompasses all existing partition theorems. We also give a natural generalization of Euler's theorem based on a special complete residue system. Furthermore, we establish interesting congruence connections between the partition function $p(n)$ and related partition functions.

math.CO↗

arXiv2014

Arithmetic Properties of Andrews' Singular Overpartitions

In a very recent work, G. E. Andrews defined the combinatorial objects which he called {\it singular overpartitions} with the goal of presenting a general theorem for overpartitions which is analogous to theorems of Rogers--Ramanujan type for ordinary partitions with restricted successive ranks. As a small part of his work, Andrews noted two congruences modulo 3 which followed from elementary generating function manipulations. In this work, we prove that Andrews' results modulo 3 are two examples of an infinite family of congruences modulo 3 which hold for that particular function. We also expand the consideration of such arithmetic properties to other functions which are part of Andrews' framework for singular overpartitions.

math.NT↗

arXiv2011

Distribution of Coefficients of Modular Forms and the Partition Function

Let $\ell\ge5$ be an odd prime and $j, s$ be positive integers. We study the distribution of the coefficients of integer and half-integral weight modular forms modulo odd positive integer $M$. As a consequence, we prove that for each integer $1\le r\le\ell^j$, $$\sharp\{1\le n\le X\ |\ p(n)\equiv r\pmod{\ell^j}\}\gg_{s,r,\ell^j}\frac{\sqrt X}{\log X}(\log\log X)^s.$$

math.NT↗

arXiv2011

Odd Multiperfect Numbers

A natural number $n$ is called {\it multiperfect} or {\it$k$-perfect} for integer $k\ge2$ if $σ(n)=kn$, where $σ(n)$ is the sum of the positive divisors of $n$. In this paper, we establish the structure theorem of odd multiperfect numbers analogous as Euler's theorem on odd perfect numbers. We prove the divisibility of the Euler part of odd multiperfect numbers and characterize the forms of odd perfect numbers $n=π^αM^2$ such that $π\equivα(\text{mod}8)$. We also present some examples to show the nonexistence of odd perfect numbers as applications.

math.NT↗

arXiv2011

Bounds for Odd $k$-Perfect Numbers

Let $k\ge2$ be an integer. A natural number $n$ is called $k$-perfect if $σ(n)=kn.$ For any integer $r\ge1$ we prove that the number of odd $k$-perfect numbers with at most $r$ distinct prime factors is bounded by $k4^{r^3}$.

math.NT↗