Searcharxiv⌕ Search

arXiv subjects

He-Xia Ni

Publications and source records attributed to He-Xia Ni.

16 recordsLinked to original sources
arXiv2026

Cyclic permutations of large subsets with polynomial values in multiplicative subgroups of finite fields

Let $f(t)\in\mathbb{Z}[t]$ be a nonconstant polynomial with nonzero discriminant and let $k\ge2$ be an integer. Inspired by the work of Alon and Bourgain, for sufficiently large prime $p\equiv1\pmod{k}$, we study cyclic orderings of subsets $A\subseteq \mathbb{F}_p$ for which $ f(a_i+a_{i+1})$ is a nonzero $k$-th power for every consecutive pair. By combining mixed character-sum estimates, Fourier analysis on $\mathbb{F}_p$, and spectral graph methods, we establish a threshold $c(p,k,f)$ such that every subset $A$ with $\#A\ge c(p,k,f)$ admits such a cyclic ordering. We also give lower and upper bounds for the optimal threshold.

math.NT↗
arXiv2026

On the Burgess bound and Pythagorean triples involving primitive roots

By combining the recent results of Pierce and Xu on the higher-dimensional Burgess bound with the classical one-dimensional Burgess bound, we prove that for any sufficiently large prime $p$, there exists a Pythagorean triple $(a_p,b_p,c_p)$ with $a_p,b_p,c_p\in\mathbb{Z}\cap(0,p)$ such that $a_pb_p/2$ is a primitive root modulo $p$. This confirms a conjecture of Z.-W. Sun for all sufficiently large primes.

math.NT↗
arXiv2026

Supercongruences Arising from Truncated Appell Series

In recent years, both Appell series and truncated Appell series have garnered significant research interest among scholars. In this paper, we investigate the congruence properties of truncated Appell series and truncated $q$-Appell series. By employing various combinatorial identities and the `creative microscoping' method introduced by Guo and Zudilin, we provide $q$-analogues for several known congruences of truncated Appell series $F_1$ and $F_2$. Meanwhile, we establish a new $q$-supercongruence related to the truncated Appell series $F_4$ and propose a related conjecture, which provides a $q$-analogue for two conjectures by Apagodu and Zeilberger.

math.NT↗
arXiv2026

A supercongruence related to Whipple's ${}_5F_4$ formula and Dwork's dash operation

We establish a parametric supercongruence related to Whipple's ${}_5F_4$ formula and Dwork's dash operation. As a typical consequence, we obtain the following result: for any prime $p\equiv3\pmod4$ and odd integer $r\geq1$, $$ \sum_{k=0}^{p^r-1}(8k+1)\frac{(\frac14)_k^3(\frac12)_k}{(1)_k^3(\frac34)_k}\equiv 3p^r+\frac{27p^{3r}}{4}H_{(p^r-3)/4}^{(2)}\pmod{p^{r+3}}, $$ where $(x)_n=x(x+1)\cdots(x+n-1)$ is the Pochhammer symbol and $H_n^{(2)}=\sum_{k=1}^n\frac{1}{k^2}$ is the $n$-th harmonic number of order $2$. This confirms a conjecture of Guo and Zhao [Forum Math. 38 (2026), 1099-1109]. Our proof rely on a new parametric WZ pair which allows us to transform the original sum to a computable form in the sense of congruence. Another essential ingredient of our proof involves the properties of Dwork's dash operation.

math.NT↗
arXiv2025

The Pell sequence and cyclotomic matrices involving squares over finite fields

In this paper, by some arithmetic properties of the Pell sequence and some $p$-adic tools, we study certain cyclotomic matrices involving squares over finite fields. For example, let $1=s_1,s_2,\cdots,s_{(q-1)/2}$ be all the nonzero squares over $\mathbb{F}_{q}$, where $q=p^f$ is an odd prime power with $q\ge7$. We prove that the matrix $$B_q((q-3)/2)=\left[\left(s_i+s_j\right)^{(q-3)/2}\right]_{2\le i,j\le (q-1)/2}$$ is a singular matrix whenever $f\ge2$. Also, for the case $q=p$, we show that $$\det B_p((p-3)/2)=0\Leftrightarrow Q_p\equiv 2\pmod{p^2\mathbb{Z}},$$ where $Q_p$ is the $p$-th term of the companion Pell sequence $\{Q_i\}_{i=0}^{\infty}$ defined by $Q_0=Q_1=2$ and $Q_{i+1}=2Q_i+Q_{i-1}$.

math.NT↗
arXiv2024

On a generalization of R. Chapman's "evil determinant"

Let $p$ be an odd prime and $x$ be an indeterminate. Recently, Z.-W. Sun proposed the following conjecture: $$\det\left[x+\left(\frac{j-i}{p}\right)\right]_{0\le i,j\le \frac{p-1}{2}}=\begin{cases} (\frac{2}{p})pb_px-a_p & \mbox{if}\ p\equiv 1\pmod4, 1 & \mbox{if}\ p\equiv 3\pmod4, \end{cases}$$ where $a_p$ and $b_p$ are rational numbers related to the fundamental unit and class number of the real quadratic field $\mathbb{Q}(\sqrt{p})$. In this paper, we confirm the above conjecture of Sun based on Vsemirnov's decomposition of Chapman's "evil determinant".

math.NT↗
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.

math.NT↗
arXiv2021

A $q$-Dwork-type generalization of Rodriguez-Villegas' supercongruences

Guo and Zudilin [Adv. Math. 346 (2019), 329--358] developed an analytical method, called `creative microscoping', to prove many supercongruences by establishing their $q$-analogues. In this paper, we apply this method to give a $q$-Dwork-type generalization of Rodriguez-Villegas' supercongruences, which was recently conjectured by Guo and Zudilin.

math.NT↗
arXiv2021

Some $q$-congruences involving central $q$-binomial coefficients

Suppose that $p$ is an odd prime and $m$ is an integer not divisible by $p$. Sun and Tauraso [Adv. in Appl. Math., 45(2010), 125--148] gave $\sum_{k=0}^{n-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=0}^{n-1}\binom{2k}{k+d}/(km^k)$ modulo $p$ for all $d=0,1, \ldots n$ and $n= p^a$, where $a$ is a positive integer. In this paper, we present some $q$-analogues of these congruences in the cases $m=2, 4$ for any positive integer $n$.

math.NT↗
arXiv2021

$q$-Supercongruences from transformation formulas

Let $Φ_{n}(q)$ denote the $n$-th cyclotomic polynomial in $q$. Recently, Guo and Schlosser [Constr. Approx. 53 (2021), 155--200] put forward the following conjecture: for an odd integer $n>1$, \begin{align*} &\sum_{k=0}^{n-1}[8k-1]\frac{(q^{-1};q^4)_k^6(q^2;q^2)_{2k}}{(q^4;q^4)_k^6(q^{-1};q^2)_{2k}}q^{8k}\notag\\ &\quad\equiv\begin{cases}0 \pmod{[n]Φ_n(q)^2}, &\text{if }n\equiv 1\pmod{4},\\[5pt] 0 \pmod{[n]},&\text{if }n\equiv 3\pmod{4}. \end{cases} \end{align*} Applying the `creative microscoping' method and several summation and transformation formulas for basic hypergeometric series and the Chinese remainder theorem for coprime polynomials, we confirm the above conjecture, as well as another similar $q$-supercongruence conjectured by Guo and Schlosser.

math.NT↗
arXiv2020

Two $q$-supercongruences from Watson's transformation

Guo and Zudilin [Adv. Math. 346 (2019), 329--358] introduced a new method called `creative microscoping', to prove many $q$-supercongruences in a unified way. In this paper, we apply this method and Watson's ${}_8ϕ_7$ transformation formula to prove two $q$-supercongruences, which were recently conjectured by Guo and Schlosser.

math.NT↗
arXiv2020

Some $q$-congruences arising from certain identities

In this paper, by constructing some identities, we prove some $q$-analogues of some congruences. For example, for any odd integer $n>1$, we show that \begin{gather*} \sum_{k=0}^{n-1} \frac{(q^{-1};q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2} q^{(n^2-1)/4} - (1+q)[n] \pmod{Φ_n(q)^2},\\ \sum_{k=0}^{n-1}\frac{(q^3;q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2} q^{(n^2-9)/4} + \frac{1+q}{q^2}[n]\pmod{Φ_n(q)^2}, \end{gather*} where the $q$-Pochhanmmer symbol is defined by $(x;q)_0=1$ and $(x;q)_k = (1-x)(1-xq)\cdots(1-xq^{k-1})$ for $k\geq1$, the $q$-integer is defined by $[n]=1+q+\cdots+q^{n-1}$ and $Φ_n(q)$ is the $n$-th cyclotomic polynomial. The $q$-congruences above confirm some recent conjectures of Gu and Guo.

math.NT↗
arXiv2020

A Remark on stress of a spatially uniform dislocation density field

In an interesting recent paper [1] (A. Acharya, Stress of a spatially uniform dislocation density field, J. Elasticity 137 (2019), 151--155), Acharya proved that the stress produced by a spatially uniform dislocation density field in a body comprising a nonlinear elastic material may fail to vanish under no loads. The class of counterexamples constructed in [1] is essentially $2$-dimensional: it works with the subgroup $\mathcal{O}(2) \oplus \langle{\bf Id}\rangle \subset \mathcal{O}(3)$. The objective of this note is to extend Acharya's result in [1] to $\mathcal{O}(3)$, subject to an additional structural assumption and less regularity requirements.

physics.class-ph↗
arXiv2018

On the almost universality of $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$

In 2013, Farhi conjectured that for each $m\geq 3$, every natural number $n$ can be represented as $\lfloor x^2/m\rfloor+\lfloor y^2/m\rfloor+\lfloor z^2/m\rfloor$ with $x,y,z\in\Z$, where $\lfloor\cdot\rfloor$ denotes the floor function. Moreover, in 2015, Sun conjectured that every natural number $n$ can be written as $\lfloor x^2/a\rfloor+\lfloor y^2/b\rfloor+\lfloor z^2/c\rfloor$ with $x,y,z\in\Z$, where $a,b,c$ are integers and $(a,b,c)\neq (1,1,1),(2,2,2)$. In this paper, with the help of congruence theta functions, we prove that for each $m\geq 3$, Farhi's conjecture is true for every sufficiently large integer $n$. And for $a,b,c\geq 5$ with $a,b,c$ are pairwisely co-prime, we also confirm Sun's conjecture for every sufficiently large integer $n$.

math.NT↗
arXiv2018

Divisibility of some binomial sums

With help of $q$-congruence, we prove the divisibility of some binomial sums. For example, for any integers $ρ,n\geq 2$, $$\sum_{k=0}^{n-1}(4k+1) \binom{2k}{k}^ρ\cdot (-4)^{ρ(n-1-k)} \equiv 0\pmod{2^{ρ-2}n\binom{2n}{n}}.$$

math.NT↗