arXiv subjects

Hai-Liang Wu

Publications and source records attributed to Hai-Liang Wu.

48 records · Page 3Linked to original sources

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

Determinants concerning Legendre symbols

The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some variants. For example, we show the following result: Let $p=a^2+4b^2$ be a prime with $a,b$ integers and $a\equiv1\pmod4$. Then for the determinant $$S(1,p):={\rm det}\bigg[\left(\frac{i^2+j^2}{p}\right)\bigg]_{1\le i,j\le \frac{p-1}{2}},$$ the number $S(1,p)/a$ is an integral square, which confirms a conjecture posed by Cohen, Sun and Vsemirnov.

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arXiv2020

Almost universal mixed sums of squares and polygonal numbers

For each integer $m\ge3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary quadratic forms to determine completely when the mixed sum $ax^2+by^2+cP_{p^k+2}(z)$ represents all but finitely many positive integers.

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arXiv2020

On the 1-3-5 conjecture and related topics

The 1-3-5 conjecture of Z.-W. Sun states that any $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $x^2+y^2+z^2+w^2$ with $w,x,y,z\in\mathbb N$ such that $x+3y+5z$ is a square. In this paper, via the theory of ternary quadratic forms and related modular forms, we study the integer version of the 1-3-5 conjecture and related weighted sums of four squares with certain linear restrictions. Here are two typical results in this paper: (i) There is a finite set $A$ of positive integers such that any sufficiently large integer not in the set $\{16^ka:\ a\in A,\ k\in\mathbb N\}$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Z$ and $x+3y+5z\in\{4^k:\ k\in\mathbb N\}$. (ii) Any positive integer can be written as $x^2+y^2+z^2+2w^2$ with $x,y,z,w\in\mathbb Z$ and $x+y+2z+2w=1$. Also, any sufficiently large integer can be written as $x^2+y^2+z^2+2w^2$ with $x,y,z,w\in\mathbb Z$ and $x+2y+3z=1$.

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arXiv2020

Squares in $\mathbb{F}_{p^2}$ and permutations involving primitive roots

Let $p=2n+1$ be an odd prime, and let $ζ_{p^2-1}$ be a primitive $(p^2-1)$-th root of unity in the algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. We let $g\in\mathbb{Z}_p[ζ_{p^2-1}]$ be a primitive root modulo $p\mathbb{Z}_p[ζ_{p^2-1}]$ with $g\equiv ζ_{p^2-1}\pmod {p\mathbb{Z}_p[ζ_{p^2-1}]}$. Let $Δ\equiv3\pmod4$ be an arbitrary quadratic non-residue modulo $p$ in $\mathbb{Z}$. By the Local Existence Theorem we know that $\mathbb{Q}_p(\sqrtΔ)=\mathbb{Q}_p(ζ_{p^2-1})$. For all $x\in\mathbb{Z}[\sqrtΔ]$ and $y\in\mathbb{Z}_p[ζ_{p^2-1}]$ we use $\bar{x}$ and $\bar{y}$ to denote the elements $x\mod p\mathbb{Z}[\sqrtΔ]$ and $y\mod p\mathbb{Z}_p[ζ_{p^2-1}]$ respectively. If we set $a_k=k+\sqrtΔ$ for $0\le k\le p-1$, then we can view the sequence $$S := \overline{a_0^2}, \cdots, \overline{a_0^2n^2}, \cdots,\overline{a_{p-1}^2}, \cdots, \overline{a_{p-1}^2n^2}\cdots, \overline{1^2}, \cdots,\overline{n^2}$$ as a permutation $σ$ of the sequence $$S^* := \overline{g^2}, \overline{g^4}, \cdots,\overline{g^{p^2-1}}.$$ We determine the sign of $σ$ completely in this paper.

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arXiv2020

Some universal quadratic sums over the integers

Let $a,b,c,d,e,f\in\mathbb N$ with $a\ge c\ge e>0$, $b\le a$ and $b\equiv a\pmod2$, $d\le c$ and $d\equiv c\pmod2$, $f\le e$ and $f\equiv e\pmod2$. If any nonnegative integer can be written as $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ with $x,y,z\in\mathbb Z$, then the ordered tuple $(a,b,c,d,e,f)$ is said to be universal over $\mathbb Z$. Recently, Z.-W. Sun found all candidates for such universal tuples over $\mathbb Z$. In this paper, we use the theory of ternary quadratic forms to show that 44 concrete tuples $(a,b,c,d,e,f)$ in Sun's list of candidates are indeed universal over $\mathbb Z$. For example, we prove the universality of $(16,4,2,0,1,1)$ over $\mathbb Z$ which is related to the form $x^2+y^2+32z^2$.

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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 $S_p(x)=\prod_{c}(x-ζ_p^c),$ where $0<c<p$ and $c$ varies over all quadratic residues modulo $p$ and $ζ_p=e^{2πi/p}$. 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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arXiv2019

Sums of squares with restrictions involving primes

The well-known Lagrange's four-square theorem states that any integer $n\in\mathbb{N}=\{0,1,2,...\}$ can be written as the sum of four squares. Recently, Z.-W. Sun investigated the representations of $n$ as $x^2+y^2+z^2+w^2$ with certain linear restrictions involving the integer variables $x,y,z,w$. In this paper, via the theory of quadratic forms, we further study the representations $n=x^2+y^2+z^2+w^2$ (resp., $n=x^2+y^2+z^2+2w^2$) with certain linear restrictions involving primes. For example, we obtain the following results: (i) Each positive integer $n>1$ can be written as $x^2+y^2+z^2+2w^2$ ($x,y,z,w\in\mathbb N$) with $x+y$ prime. (ii) Every positive integer can be written as $x^2+y^2+z^2+2w^2$ ($x,y,z,w\in\mathbb N$) with $x+2y$ prime. (iii) Let $k$ be any positive integer, and let $d\ge 2^{k-1}$ be a positive odd integer with $4d^2+1$ prime. Then any sufficiently large integer can be written as $x^2+y^2+z^2+2w^2$ $(x,y,z,w\in\mathbb N)$ with $x+2dy=p^k$ for some prime $p$.

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arXiv2019

On some determinants involving cyclotomic units

For each odd prime $p$, let $ζ_p$ denote a primitive $p$-th root of unity. In this paper, we study the determinants of some matrices with cyclotomic unit entries. For instance, we show that when $p\equiv 3\pmod4$ and $p>3$ the determinant of the matrix $$\frac{1-ζ_p^{j^2k^2}}{1-ζ_p^{j^2}}$_{1\le j,k\le (p-1)/2}$ can be written as $(-1)^{\frac{h(-p)+1}{2}}(a_p+b_pi\sqrt{p})$ with $a_p,b_p\in\frac12\Z$ and $$\begin{cases}ν_p(a_p)=ν_p(b_p)=\frac{p-3}{8}&\mbox{if}\ p\equiv 3\pmod8, \\ν_p(a_p)=ν_p(b_p)+1=\frac{p+1}{8}&\mbox{if}\ p\equiv 7\pmod8,\end{cases}$$ where $ν_p(x)$ denotes the $p$-adic order of a $p$-adic integer $x$, and $h(-p)$ denotes the class number of the field $\Q(\sqrt{-p})$. Meanwhile, let $(\frac{\cdot}{p})$ denote the Legendre symbol. We have $$2^{\frac{p+1}{2}}a_pb_p=(-1){^\frac{p+1}{4}}p^{\frac{p-3}{4}}\det [S(p)],$$ and $$2^{\frac{p-1}{2}}(a_p^2-pb_p^2)=\frac{p-1}{2}(-p)^{\frac{p-3}{4}}\det [S(p)],$$ where $\det [S(p)]$ is the determinant of the $\frac{p-1}{2}$ by $\frac{p-1}{2}$ matrix $S(p)$ with entries $S(p)_{j,k}=(\frac{j^2+k^2}{p})$ for any $1\le j,k\le (p-1)/2$.

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arXiv2019

Applications of Lerch's theorem and permutations concerning quadratic residues

Let $p$ be an odd prime. For each integer $a$ with $p\nmid a$, the famous Zolotarev's Lemma says that the Legendre symbol $(\frac{a}{p})$ is the sign of the permutation of $\Z/p\Z$ induced by multiplication by $a$. The extension of Zolotarev's result to the case of odd integers was shown by Frobenius. After that, Lerch extended these to all positive integers. In this paper we explore some applications of Lerch's result. For instance, we study permutations involving arbitrary $k$-th power residue modulo $p$ and primitive roots of a power of $p$. Finally, we discuss some permutation problems concerning quadratic residues modulo $p$. In particular, we confirm some conjectures posed by Sun.

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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 $n$.

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arXiv2018

On the sums of three generalized polygonal numbers

For each natural number $m\ge 3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. In this paper, with the help of the congruence theta function, we establish conditions on $a$, $b$, $c$ for which the sum $P_a(x)+P_b(y)+P_c(z)$ represents all but finitely many positive integers.

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