arXiv subjects

Zhi-Wei Sun

Publications and source records attributed to Zhi-Wei Sun.

At least 109 records · Page 6Linked to original sources

arXiv2020

Universal sums of three quadratic polynomials

Let $a,b,c,d,e$ and $f$ be integers with $a\ge c\ge e>0$, $b>-a$ and $b\equiv a\pmod2$, $d>-c$ and $d\equiv c\pmod 2$, $f>-e$ and $f\equiv e\pmod2$. Suppose that $b\ge d$ if $a=c$, and $d\ge f$ if $c=e$. When $b(a-b)$, $d(c-d)$ and $f(e-f)$ are not all zero, we prove that if each $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ with $x,y,z\in\mathbb N$ then the tuple $(a,b,c,d,e,f)$ must be on our list of $473$ candidates, and show that 56 of them meet our purpose. When $b\in[0,a)$, $d\in[0,c)$ and $f\in[0,e)$, we investigate the universal tuples $(a,b,c,d,e,f)$ over $\mathbb Z$ for which any $n\in\mathbb N$ can be written $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ with $x,y,z\in\mathbb Z$, and show that there are totally 12082 such candidates some of which are proved to be universal tuples over $\mathbb Z$. For example, we show that any $n\in\mathbb N$ can be written as $x(x+1)/2+y(3y+1)/2+z(5z+1)/2$ with $x,y,z\in\mathbb Z$, and conjecture that each $n\in\mathbb N$ can be written as $x(x+1)/2+y(3y+1)/2+z(5z+1)/2$ with $x,y,z\in\mathbb N$.

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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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arXiv2019

Two $q$-analogues of Euler's formula $ζ(2)=π^2/6$

It is well known that $ζ(2)=π^2/6$ as discovered by Euler. In this paper we present the following two $q$-analogues of this celebrated formula: $$\sum_{k=0}^\infty\frac{q^k(1+q^{2k+1})}{(1-q^{2k+1})^2}=\prod_{n=1}^\infty\frac{(1-q^{2n})^4}{(1-q^{2n-1})^4}$$ and $$\sum_{k=0}^\infty\frac{q^{2k-\lfloor(-1)^kk/2\rfloor}}{(1-q^{2k+1})^2} =\prod_{n=1}^\infty\frac{(1-q^{2n})^2(1-q^{4n})^2}{(1-q^{2n-1})^2(1-q^{4n-2})^2},$$ where $q$ is any complex number with $|q|<1$. We also give a $q$-analogue of the identity $ζ(4)=π^4/90$, and pose a problem on $q$-analogues of Euler's formula for $ζ(2m)\ (m=3,4,\ldots)$.

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arXiv2019

$p$-adic analogues of hypergeometric identities and their applications

In this paper, we confirm several conjectures posed by Sun recently; for example, we prove that for any odd prime $p$ we have $$ \sum_{k=0}^{p-1}A_k\equiv\begin{cases}4x^2-2p\pmod{p^2}\quad&\text{if $p=x^2+2y^2\ (x,y\in\mathbb{Z})$},\\ 0\pmod{p^2}\quad&\text{if $p\equiv5,7\pmod{8}$},\end{cases} $$ where $A_n:=\sum_{k=0}^n\binom{n+k}{k}^2\binom{n}{k}^2$ are the Apéry numbers.

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arXiv2019

Restricted sums of four squares

We refine Lagrange's four-square theorem in new ways by imposing some restrictions involving powers of two (including $1$). For example, we show that each $n=1,2,3,\ldots$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb N=\{0,1,2,\ldots\})$ with $|x+y-z|\in\{4^k:\ k\in\mathbb N\}$ (or $|2x-y|\in\{4^k:\ k\in\mathbb N\}$, or $x+y-z\in\{\pm 8^k:\ k\in\mathbb N\}\cup\{0\}\subseteq\{t^3:\ t\in\mathbb Z\}$), and that we can write any positive integer as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+2z$ (or $x+2y+2z$) a power of four. We also prove that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+2w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+z+w$ a square (or a cube). In addition, we pose some open conjectures for further research; for example, we conjecture that any integer $n>1$ can be written as $a^2+b^2+3^c+5^d$ with $a,b,c,d\in\mathbb N$.

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arXiv2019

Consecutive primes and Legendre symbols

Let $m$ be any positive integer and let $δ_1,δ_2\in\{1,-1\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\le C_m$ such that $$\left(\frac{p_{n+i}}{p_{n+j}}\right)=δ_1\ \quad\text{and}\ \quad\left(\frac{p_{n+j}}{p_{n+i}}\right)=δ_2$$ for all $0\le i<j\le m$, where $p_k$ denotes the $k$-th prime, and $(\frac {\cdot}p)$ denotes the Legendre symbol for any odd prime $p$. We also prove that under the Generalized Riemann Hypothesis there are infinitely many positive integers $n$ such that $p_{n+i}$ is a primitive root modulo $p_{n+j}$ for any distinct $i$ and $j$ among $0,1,\ldots,m$.

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arXiv2019

On practical numbers of some special forms

In this paper we study practical numbers of some special forms. For any integers $b\ge0$ and $c>0$, we show that if $n^2+bn+c$ is practical for some integer $n>1$, then there are infinitely many nonnegative integers $n$ with $n^2+bn+c$ practical. We also prove that there are infinitely many practical numbers of the form $q^4+2$ with $q$ practical, and that there are infinitely many practical Pythagorean triples $(a,b,c)$ with $\gcd(a,b,c)=6$ (or $\gcd(a,b,c)=4$).

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arXiv2019

Quadratic residues and related permutations and identities

Let $p$ be an odd prime. In this paper we investigate quadratic residues modulo $p$ and related permutations, congruences and identities. If $a_1<\ldots<a_{(p-1)/2}$ are all the quadratic residues modulo $p$ among $1,\ldots,p-1$, then the list $\{1^2\}_p,\ldots,\{((p-1)/2)^2\}_p$ (with $\{k\}_p$ the least nonnegative residue of $k$ modulo $p$) is a permutation of $a_1,\ldots,a_{(p-1)/2}$, and we show that the sign of this permutation is $1$ or $(-1)^{(h(-p)+1)/2}$ according as $p\equiv3\pmod 8$ or $p\equiv7\pmod 8$, where $h(-p)$ is the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$. To achieve this, we evaluate the product $\prod_{1\le j<k\le(p-1)/2}(\cotπj^2/p-\cotπk^2/p)$ via Dirichlet's class number formula and Galois theory. We also obtain some new identities for the sine and cosine functions; for example, we determine the exact value of $$\prod_{1\le j<k\le p-1}\cosπ\frac{aj^2+bjk+ck^2}p$$ for any $a,b,c\in\mathbb Z$ with $ac(a+b+c)\not\equiv0\pmod p$.

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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 with Legendre symbol entries

In this paper we mainly focus on some determinants with Legendre symbol entries. Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. We show that $(\frac{-S(d,p)}p)=1$ for any $d\in\mathbb Z$ with $(\frac dp)=1$, and that $$\left(\frac{W_p}p\right)=\begin{cases}(-1)^{|\{0<k<\frac p4:\ (\frac kp)=-1\}|}&\text{if}\ p\equiv1\pmod4, \\(-1)^{\lfloor(p+1)/8\rfloor}&\text{if}\ p\equiv3\pmod4,\end{cases}$$ where $$S(d,p)=\det\left[\left(\frac{i^2+dj^2}p\right)\right]_{1\le i,j\le(p-1)/2}$$ and $$W_p=\det\left[\left(\frac{i^2-((p-1)/2)!j}p\right)\right]_{0\le i,j\le(p-1)/2}.$$ We also pose some conjectures on determinants, one of which states that $(-1)^{\lfloor(p+1)/8\rfloor}W_p$ is a square when $p\equiv 3\pmod4$.

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arXiv2019

On $q$-analogues of some series for $π$ and $π^2$

We obtain a new $q$-analogue of the classical Leibniz series $\sum_{k=0}^\infty(-1)^k/(2k+1)=π/4$, namely \begin{equation*} \sum_{k=0}^\infty\frac{(-1)^kq^{k(k+3)/2}}{1-q^{2k+1}}=\frac{(q^2;q^2)_{\infty}(q^8;q^8)_{\infty}}{(q;q^2)_{\infty}(q^4;q^8)_{\infty}}, \end{equation*} where $q$ is a complex number with $|q|<1$. We also show that the Zeilberger-type series $\sum_{k=1}^\infty(3k-1)16^k/(k\binom{2k}k)^3=π^2/2$ has two $q$-analogues with $|q|<1$, one of which is $$\sum_{n=0}^\infty q^{n(n+1)/2} \frac {1-q^{3n+2}} {1-q} \cdot\frac{(q;q)_n^3 (-q;q)_n}{(q^3;q^2)_{n}^3} = (1-q)^2 \frac{(q^2;q^2)^4_\infty}{(q;q^2)^4_\infty}.$$

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arXiv2018

Two new kinds of numbers and related divisibility results

We mainly introduce two new kinds of numbers given by $$R_n=\sum_{k=0}^n\binom nk\binom{n+k}k\frac1{2k-1}\quad\ (n=0,1,2,...)$$ and $$S_n=\sum_{k=0}^n\binom nk^2\binom{2k}k(2k+1)\quad\ (n=0,1,2,...).$$ We find that such numbers have many interesting arithmetic properties. For example, if $p\equiv1\pmod 4$ is a prime with $p=x^2+y^2$ (where $x\equiv1\pmod 4$ and $y\equiv0\pmod 2$), then $$R_{(p-1)/2}\equiv p-(-1)^{(p-1)/4}2x\pmod{p^2}.$$ Also, $$\frac1{n^2}\sum_{k=0}^{n-1}S_k\in\mathbb Z\ \ {and}\ \ \frac1n\sum_{k=0}^{n-1}S_k(x)\in\mathbb Z[x]\quad\text{for all}\ n=1,2,3,...,$$ where $S_k(x)=\sum_{j=0}^k\binom kj^2\binom{2j}j(2j+1)x^j$. For any positive integers $a$ and $n$, we show that, somewhat surprisingly, $$\frac1{n^2}\sum_{k=0}^{n-1}(2k+1)\binom{n-1}k^a\binom{-n-1}k^a\in\mathbb Z\ \ {and} \ \ \frac 1n\sum_{k=0}^{n-1}\frac{\binom{n-1}k^a\binom{-n-1}k^a}{4k^2-1}\in\mathbb Z.$$ We also solve a conjecture of V.J.W. Guo and J. Zeng, and pose several conjectures for further research.

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arXiv2018

Hankel-type determinants for some combinatorial sequences

In this paper we confirm several conjectures of Z.-W. Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Apéry numbers. For any nonnegative integer $n$, define \begin{gather*}f_n:=\sum_{k=0}^n\binom nk^3,\ D_n:=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2(n-k)}{n-k}, b_n:=\sum_{k=0}^n\binom nk^2\binom{n+k}k,\ A_n:=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2. \end{gather*} For $n=0,1,2,\ldots$, we show that $6^{-n}|f_{i+j}|_{0\leq i,j\leq n}$ and $12^{-n}|D_{i+j}|_{0\le i,j\le n}$ are positive odd integers, and $10^{-n}|b_{i+j}|_{0\leq i,j\leq n}$ and $24^{-n}|A_{i+j}|_{0\leq i,j\leq n}$ are always integers.

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arXiv2018

Some variants of Lagrange's four squares theorem

Lagrange's four squares theorem is a classical theorem in number theory. Recently, Z.-W. Sun found that it can be further refined in various ways. In this paper we study some conjectures of Sun and obtain various refinements of Lagrange's theorem. We show that any nonnegative integer can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+z+w$ (or $x+y+z+2w$, or $x+2y+3z+w$) a square (or a cube). Also, every $n=0,1,2,\ldots$ can be represented by $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+3z$ (or $x+2y+3z$) a square (or a cube), and each $n=0,1,2,\ldots$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $(10w+5x)^2+(12y+36z)^2$ (or $x^2y^2+9y^2z^2+9z^2x^2$) a square. We also provide an advance on the 1-3-5 conjecture of Sun. Our main results are proved by a new approach involving Euler's four-square identity

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arXiv2018

On sums and products in a field

In this paper we study sums and products in a field. Let $F$ be a field with ${\rm ch}(F)\not=2$, where ${\rm ch}(F)$ is the characteristic of $F$. For any integer $k\ge4$, we show that each $x\in F$ can be written as $a_1+\ldots+a_k$ with $a_1,\ldots,a_k\in F$ and $a_1\ldots a_k=1$ if ${\rm ch}(F)\not=3$, and that for any $α\in F\setminus\{0\}$ we can write each $x\in F$ as $a_1\ldots a_k$ with $a_1,\ldots,a_k\in F$ and $a_1+\ldots+a_k=α$. We also prove that for any $x\in F$ and $k\in\{2,3,\ldots\}$ there are $a_1,\ldots,a_{2k}\in F$ such that $a_1+\ldots+a_{2k}=x=a_1\ldots a_{2k}$.

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arXiv2017

Some new series for $1/π$ and related congruences

In this paper we prove some new series for $1/π$ as well as related congruences. We also raise several new kinds of series for $1/π$ and present some related conjectural congruences involving representations of primes by binary quadratic forms.

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arXiv2017

On a permutation problem for finite abelian groups

Let $G$ be a finite additive abelian group with exponent $n>1$, and let $a_1,\ldots,a_{n-1}\in G$. We show that there is a permutation $σ\in S_{n-1}$ such that all the elements $sa_{σ(s)}\ (s=1,\ldots,n-1)$ are nonzero if and only if $$\left|\left\{1\le s<n:\ \frac{n}{d}a_s\ne 0\right\}\right|\ge d-1\ \ \textrm{ for every positive divisor }\ d\ \textrm{ of }\ n.$$ When $G$ is the cyclic group $\mathbb Z/n\mathbb Z$, this confirms a conjecture of Z.-W. Sun.

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arXiv2017

Conjectures on representations involving primes

We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists $k\in\{0,\ldots,n\}$ such that $n+k$ and $n+k^2$ are both prime. (ii) Each integer $n>1$ can be written as $x+y$ with $x,y\in\{1,2,3,\ldots\}$ such that $x+ny$ and $x^2+ny^2$ are both prime. (iii) For any rational number $r>0$, there are distinct primes $q_1,\ldots,q_k$ with $r=\sum_{j=1}^k1/(q_j-1)$. (iv) Every $n=4,5,\ldots$ can be written as $p+q$, where $p$ is a prime with $p-1$ and $p+1$ both practical, and $q$ is either prime or practical. (v) Any positive rational number can be written as $m/n$, where $m$ and $n$ are positive integers with $p_m+p_n$ a square (or $π(m)π(n)$ a positive square), $p_k$ is the $k$-th prime and $π(x)$ is the prime-counting function.

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