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Yue-Feng She

Publications and source records attributed to Yue-Feng She.

16 recordsLinked to original sources
arXiv2026

Proof of a conjecture on permutations

Given a positive integer $n$, define a function on the symmetric group $S_n$ by $$F(\tau) = \sum_{k=1}^{n}k^2\tau(k)^2.$$ Motivated by a conjecture of Zhi-Wei Sun, we investigate the residue classes attained by $F(\tau)$ modulo $2n+1$. We prove that for every integer $n>4$, the set $\{F(\tau):\tau\in S_n\}$ contains a complete residue system modulo $2n+1$. The proof is based on a family of involutions whose values are controlled by subset sums of squares.

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arXiv2026

Representations of positive integers by three almost-prime squares

Let $P_r$ denote an integer with at most $r$ prime factors, counted with multiplicity. It is known that every sufficiently large integer $N$ satisfying $N \equiv 3 \pmod{24}$ and $5 \nmid N$, can be written in the form $N= x_1^2+x_2^2+x_3^2$ where $x_1,x_2,x_3$ are integers. In this paper, we prove that the above representation in the following two different forms (i) $x_1x_2x_3$ is a $P_{67}$-number; (ii) each $x_i$ is a $P_{27}$-number. This result improves on the previous result of Waibel\cite{Wa}, in which $P_{72}$ was obtained in place of $P_{67}$. The proofs combine the higher-dimensional sieve, a Richert-type weighted sieve method introduced by Cai \cite{Cai} with a Bombieri-Vinogradov type result given by Waibel\cite{Wa}. Applying the same method in a one dimensional sieve setting, we also show that every sufficiently large $N$ not of the form $4^k(8l+7)$ can be written in the form \[ N = x^{2} + y^{2} + (2^{a} z)^{2}, \] where $x,y,a,z$ are non-negative integers and $z$ is a $P_{18}$-number. This improves upon a result of Banerjee \cite{Ban} who obtained $P_{118}$ in place of $P_{18}$.

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arXiv2025

On restricted sums of four squares and Zhi-Wei Sun's $x+24y$ conjecture

In this paper, by using the arithmetic theory of ternary quadratic forms, we study some refinements on Lagrange's four-square theorem. For example, given positive integers $a,b$ satisfying some algebraic conditions and a positive integer $C\ge3$, we will show that for any sufficiently large integer $n$ with $\ord_2(n)\le C$, there exist non-negative integers $x,y,z,w$ such that $$ \begin{cases} x^2+y^2+z^2+w^2=n, ax+by\in\mathcal{S}, \end{cases} $$ where $\mathcal{S}$ is the set of all squares over $\mathbb{Z}$. In particular, we obtain some progress on Zhi-Wei Sun's $x+24y$ conjecture.

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arXiv2024

Some determinants involving binary forms

In this paper, we study arithmetic properties of certain determinants involving powers of $i^2+cij+dj^2$, where $c$ and $d$ are integers. For example, for any odd integer $n>1$ with $(\frac dn)=-1$ we prove that $\det [ (\frac{i^2+cij+dj^2}{n})]_{0\le i,j\le n-1}$ is divisible by $\varphi(n)^2$, where $(\frac{\cdot}{n})$ is the Jacobi symbol and $\varphi$ is Euler's totient function. This confirms a previous conjecture of the second author.

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arXiv2022

A novel permanent identity with applications

Let $n$ be a positive integer, and define the rational function $S(x_1,\ldots,x_{2n})$ as the permanent of the matrix $[x_{j,k}]_{1\le j,k\le 2n}$, where $$x_{j,k}=\begin{cases}(x_j+x_k)/(x_j-x_k)&\text{if}\ j\not=k,\\1&\text{if}\ j=k.\end{cases}$$ We give an explicit formula for $S(x_1,\ldots,x_{2n})$ which has the following consequence: If one of the variables $x_1,\ldots,x_{2n}$ takes zero, then $S(x_1,\ldots,x_{2n})$ vanishes, i.e., $$\sum_{τ\in S_{2n}}\prod_{j=1\atop τ(j)\not=j}^{2n}\frac{x_j+x_{τ(j)}}{x_j-x_{τ(j)}}=0,$$ where we view an empty product $\prod_{i\in\emptyset}a_i$ as $1$. As an application, we show that if $ζ$ is a primitive $2n$-th root of unity then $$\sum_{τ\in S_{2n}}\prod_{j=1\atop τ(j)\not=j}^{2n}\frac{1+ζ^{j-τ(j)}}{1-ζ^{j-τ(j)}}=((2n-1)!!)^2$$ as conjectured by Z.-W. Sun.

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arXiv2022

Additive decompositions of cubes in finite fields

Let $p\equiv1\pmod3$ be a prime . We study several topics on additive decompositions concerning the set $C_p$ of all non-zero cubes in the finite field of $p$ elements. For example, we show that when $p>184291$ , the set $C_p$ has no decomposition of the form $C_p=A+B+C$ with $|A|,|B|,|C|\ge2$.

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arXiv2022

A conjecture of Zhi-Wei Sun on determinants over finite fields

In this paper, we study certain determinants over finite fields. Let $\mathbb{F}_q$ be the finite field of $q$ elements and let $a_1,a_2,\cdots,a_{q-1}$ be all nonzero elements of $\mathbb{F}_q$. Let $T_q=\left[\frac{1}{a_i^2-a_ia_j+a_j^2}\right]_{1\le i,j\le q-1}$ be a matrix over $\mathbb{F}_q$. We obtain the explicit value of $\det T_q$. Also, as a consequence of our result, we confirm a conjecture posed by Zhi-Wei Sun.

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arXiv2021

Numbers represented by restricted sums of four squares

In this paper, we prove some results of restricted sums of four squares using arithmetic of quaternions in the ring of Lipschitz integers. For example, we show that every nonnegative integer $n$ can be written as $x^{2}+y^{2}+z^{2}+t^{2}$ where $x,y,z,t$ are integers and $x+y+2z+2t$ is a square or a cube.

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arXiv2021

Cubes in finite fields and related permutations

Let $p=3n+1$ be a prime with $n\in\mathbb{N}=\{0,1,\cdots\}$, and let $g\in\mathbb{Z}$ be a primitive root modulo $p$. Let $0<a_1<\cdots<a_n<p$ be all the cubic residues modulo $p$ in the interval $(0,p)$. Then clearly the sequence $$a_1\ {\rm mod}\ p,\ a_2\ {\rm mod}\ p,\cdots, a_n\ {\rm mod}\ p$$ is a permutation $s_p(g)$ of the sequence $$g^3\ {\rm mod}\ p,\ g^6\ {\rm mod}\ p,\cdots, g^{3n}\ {\rm mod}\ p.$$ In this paper, we shall determine the sign of this permutation.

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arXiv2021

Primitive elements and $k$-th powers in finite fields

Let $\mathbb{F}_q$ be the finite field of $q$ elements, and let $k\mid q-1$ be a positive integer. Let $f(x)=ax^2+bx+c$ be a quadratic polynomial in $\mathbb{F}_q[x]$ with $b^2-4ac\ne0$. In this paper, we show that if $q>\max\{e^{e^3},(2k)^6\}$, then there is a primitive element $g$ of $\mathbb{F}_q$ such that $f(g)\in\mathbb{F}_q^{\times k}=\{x^k: x\in\mathbb{F}_q\setminus\{0\}\}$. Moreover, we shall confirm a conjecture posed by Sun.

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arXiv2020

Sums of four squares with a certain restriction

In 2016, while studying restricted sums of integral squares, Sun posed the following conjecture: Every positive integer $n$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb{N}=\{0,1,\cdots\})$ with $x+3y$ a square. Meanwhile, he also conjectured that for each positive integer $n$ there exist integers $x,y,z,w$ such that $n=x^2+y^2+z^2+w^2$ and $x+3y\in\{4^k:k\in\mathbb{N}\}$. In this paper, we confirm these conjectures via some arithmetic theory of ternary quadratic forms.

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arXiv2020

On a polynomial involving roots of unity and its applications

Let $p>3$ be a prime. Gauss first introduced the polynomial $S_p(x)=\prod_{c}(x-\zeta_p^c),$ where $0<c<p$ and $c$ varies over all quadratic residues modulo $p$ and $\zeta_p=e^{2\pi i/p}$. Later Dirichlet investigated this polynomial and used this to solve the problems involving the Pell equations. Recently, Z.-W Sun studied some trigonometric identities involving this polynomial. In this paper, we generalized their results. As applications of our result, we extend S. Chowla's result on the congruence concerning the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and give an equivalent form of the extended Ankeny-Artin-Chowla conjecture.

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