arXiv subjects
Publications and source records attributed to Zhi-Wei Sun.
In this paper we show that there is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{52})=0$ (with integer coefficients) over the Gaussian ring $\mathbb Z[i]$ is solvable.
Hilbert's Tenth Problem (HTP) asks for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring $\mathbb Z$ of the integers. This was finally solved by Matiyasevich negatively in 1970. In this paper we obtain some further results on HTP over $\mathbb Z$. We prove that there is no algorithm to determine for any $P(z_1,\ldots,z_9)\in\mathbb Z[z_1,\ldots,z_9]$ whether the equation $P(z_1,\ldots,z_9)=0$ has integral solutions with $z_9\ge0$. Consequently, there is no algorithm to test whether an arbitrary polynomial Diophantine equation $P(z_1,\ldots,z_{11})=0$ (with integer coefficients) in 11 unknowns has integral solutions, which provides the best record on the original HTP over $\mathbb Z$. We also prove that there is no algorithm to test for any $P(z_1,\ldots,z_{17})\in\mathbb Z[z_1,\ldots,z_{17}]$ whether $P(z_1^2,\ldots,z_{17}^2)=0$ has integral solutions, and that there is a polynomial $Q(z_1,\ldots,z_{20})\in\mathbb Z[z_1,\ldots,z_{20}]$ such that $$\{Q(z_1^2,\ldots,z_{20}^2):\ z_1,\ldots,z_{20}\in\mathbb Z\}\cap\{0,1,2,\ldots\}$$ coincides with the set of all primes.
For $A,B\in\mathbb Z$, the Lucas sequence $u_n(A,B)\ (n=0,1,2,\ldots)$ are defined by $u_0(A,B)=0$, $u_1(A,B)=1$, and $u_{n+1}(A,B) = Au_n(A,B)-Bu_{n-1}(A,B)$ $(n=1,2,3,\ldots).$ For any odd prime $p$ and positive integer $n$, we establish the new result $$\frac{u_{pn}(A,B) - (\frac{A^2-4B}p) u_n(A,B)}{pn} \in \mathbb Z_p,$$ where $(\frac{\cdot}p)$ is the Legendre symbol and $\mathbb Z_p$ is the ring of $p$-adic integers. Let $p$ be an odd prime and let $n$ be a positive integer. For any integer $m\not\equiv0\pmod p$, we show that $$\frac1{pn}\bigg(\sum_{k=0}^{pn-1} \frac{\binom{2k}k}{m^k} -\left(\fracΔp\right) \sum_{r=0}^{n-1}\frac{\binom{2r}r}{m^r}\bigg)\in\mathbb Z_p$$ and furthermore $$\frac1n\bigg(\sum_{k=0}^{pn-1} \frac{\binom{2k}k}{m^k} -\left(\fracΔp\right) \sum_{r=0}^{n-1}\frac{\binom{2r}r}{m^r}\bigg)\equiv \frac{\binom{2n-1}{n-1}}{m^{n-1}} u_{p-(\fracΔp)}(m-2,1) \pmod{p^2}$$ where $Δ=m(m-4)$. We also pose some conjectures for further research.
In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer $n>3$ divides the determinant $$\left|(i^2+dj^2)\left(\frac{i^2+dj^2}n\right)\right|_{0\le i,j\le (n-1)/2},$$ where $d$ is any integer and $(\frac{\cdot}n)$ is the Jacobi symbol. Then we prove some divisibility results concerning $|(i+dj)^n|_{0\le i,j\le n-1}$ and $|(i^2+dj^2)^n|_{0\le i,j\le n-1}$, where $d\not=0$ and $n>2$ are integers. Finally, for any odd prime $p$ and integers $c$ and $d$ with $p\nmid cd$, we determine completely the Legendre symbol $(\frac{S_c(d,p)}p)$, where $S_c(d,p):=|(\frac{i^2+dj^2+c}p)|_{1\le i,j\le(p-1)/2}$.
In this note, we show that each positive rational number can be written as $φ(m^2)/φ(n^2)$, where $φ$ is Euler's totient function and $m$ and $n$ are positive integers.
Let $p$ be any prime and let $a$ and $n$ be positive integers with $p\nmid n$. We show that $$\sum_{k=1}^{p^a-1}\frac{B_k}{(-n)^k}\equiv a(-1)^{n-1}D_{n-1}\pmod {p},$$ where $B_0,B_1,\ldots$ are the Bell numbers and $D_0,D_1,\ldots$ are the derangement numbers. This extends a result of Sun and Zagier published in 2011. Furthermore, we prove that $$(-x)^n\sum_{k=1}^{p^a-1}\frac{B_k(x)}{(-n)^k}\equiv -\sum_{r=1}^ax^{p^r}\sum_{k=0}^{n-1}\frac{(n-1)!}{k!}(-x)^k\pmod{p\mathbb Z_p[x]},$$ where $B_k(x)=\sum_{l=0}^kS(k,l)x^l$ is the Bell polynomial of degree $k$ with $S(k,l)\ (0\le l\le k)$ the Stirling numbers of the second kind, and $\mathbb Z_p$ is the ring of all $p$-adic integers.
In this paper we study some products related to quadratic residues and quartic residues modulo primes. Let $p$ be an odd prime and let $A$ be any integer. We mainly determine completely the product $$f_p(A):=\prod_{1\le i,j\le(p-1)/2\atop p\nmid i^2-Aij-j^2}(i^2-Aij-j^2)$$ modulo $p$; for example, if $p\equiv1\pmod4$ then $$f_p(A)\equiv\begin{cases}-(A^2+4)^{(p-1)/4}\pmod p&\text{if}\ (\frac{A^2+4}p)=1, \\(-A^2-4)^{(p-1)/4}\pmod p&\text{if}\ (\frac{A^2+4}p)=-1,\end{cases}$$ where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also determine $$\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i^2+5ij+2j^2}\left(2i^2+5ij+2j^2\right) \ \text{and}\ \prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i^2-5ij+2j^2}\left(2i^2-5ij+2j^2\right)$$ modulo $p$.
In this paper we establish connections between covers of $\mathbb Z$ by residue classes and subset sums in a field. Suppose that $A_0=\{a_s(n_s)\}_{s=0}^k$ covers each integer at least $p$ times with the residue class $a_0(n_0)=a_0+n_0\mathbb Z$ irredundant, where $p$ is a prime not dividing any of $n_1,\ldots,n_k$. Let $m_1,\ldots,m_k\in\mathbb Z$ be relatively prime to $n_1,\ldots,n_k$ respectively. For any $c,c_1,\ldots,c_k\in\mathbb Z/p\mathbb Z$ with $c_1\cdots c_k\not=0$, we show that the set $$\bigg\{\bigg\{\sum_{s\in I}\frac{m_s}{n_s}\bigg\}:\, I\subseteq\{1,\ldots,k\} \ \mbox{and}\ \sum_{s\in I}c_s=c\bigg\}$$ contains an arithmetic progression of length $n_0$ with common difference $1/n_0$, where $\{x\}$ denotes the fractional part of a real number $x$.
Via symbolic computation we deduce 97 new type series for powers of $π$ related to Ramanujan-type series. Here are three typical examples: $$\sum_{k=0}^\infty \frac{P(k) \binom{2k}k\binom{3k}k \binom{6k}{3k}}{(k+1)(2k-1)(6k-1)(-640320)^{3k}} =\frac{18\times557403^3\sqrt{10005}}{5π}$$ with \begin{align*}P(k) = &637379600041024803108 k^2 + 657229991696087780968 k \\&+ 19850391655004126179, \end{align*} $$\sum_{k=1}^\infty \frac{(3k+1)16^k}{(2k+1)^2k^3\binom{2k}k^3} = \frac{π^2-8}2,$$ and $$\sum_{n=0}^\infty\frac{3n+1}{(-100)^n} \sum_{k=0}^n{n\choose k}^2T_k(1,25)T_{n-k}(1,25) = \frac{25}{8π},$$ where the generalized central trinomial coefficient $T_k(b,c)$ denotes the coefficient of $x^k$ in the expansion of $(x^2+bx+c)^k$. We also formulate a general characterization of rational Ramanujan-type series for $1/π$ via congruences, and pose 117 new conjectural series for powers of $π$ via looking for corresponding congruences. For example, we conjecture that $$\sum_{k=0}^\infty\frac{39480k+7321}{(-29700)^k}T_k(14,1)T_k(11,-11)^2=\frac{6795\sqrt5}π.$$ Eighteen of the new series in this paper involve some imaginary quadratic fields with class number $8$.
In this paper we establish some congruences involving the Apéry numbers $β_{n}=\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}$ $(n=0,1,2,\ldots)$. For example, we show that $$\sum_{k=0}^{n-1}(11k^2+13k+4)β_k\equiv0\pmod{2n^2}$$ for any positive integer $n$, and $$\sum_{k=0}^{p-1}(11k^2+13k+4)β_k\equiv 4p^2+4p^7B_{p-5}\pmod{p^8}$$ for any prime $p>3$, where $B_{p-5}$ is the $(p-5)$th Bernoulli number. We also present certain relations between congruence properties of the two kinds of Apery numbers, $β_n$ and $A_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2$.
We obtain $q$-analogues of several series for powers of $π$. For example, the identity $$\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^3}=\frac{π^3}{32}$$ has the following $q$-analogue: \begin{equation*} \sum_{k=0}^\infty(-1)^k\frac{q^{2k}(1+q^{2k+1})}{(1-q^{2k+1})^3}=\frac{(q^2;q^4)_{\infty}^2(q^4;q^4)_{\infty}^6} {(q;q^2)_{\infty}^4}, \end{equation*} where $q$ is any complex number with $|q|<1$. We also give $q$-analogues of four new series for powers of $π$ found by the second author.
An open conjecture of Z.-W. Sun states that for any integer $n>1$ there is a positive integer $k\le n$ such that $π(kn)$ is prime, where $π(x)$ denotes the number of primes not exceeding $x$. In this paper, we show that for any positive integer $n$ the set $\{π(kn):\ k=1,2,3,\ldots\}$ contains infinitely many $P_2$-numbers which are products of at most two primes. We also prove that under the Bateman--Horn conjecture the set $\{π(4k):\ k=1,2,3,\ldots\}$ contains infinitely many primes.
The pentagonal numbers are the integers given by $p_5(n)=n(3n-1)/2\ (n=0,1,2,\ldots)$. Let $(b,c,d)$ be one of the triples $(1,1,2),(1,2,3),(1,2,6)$ and $(2,3,4)$. We show that each $n=0,1,2,\ldots$ can be written as $w+bx+cy+dz$ with $w,x,y,z$ pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integer is a sum of five pentagonal numbers two of which are equal; this refines a classical result of Cauchy claimed by Fermat.
In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer $n\equiv3\pmod4$, we show that $$(6,1)_n=[6,1]_n=(3,2)_n=[3,2]_n=0$$ and $$(4,2)_n=(8,8)_n=(3,3)_n=(21,112)_n=0$$ as conjectured by Sun, where $$(c,d)_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{1\le i,j\le n-1}$$ and $$[c,d]_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{0\le i,j\le n-1}$$ with $(\frac{\cdot}n)$ the Jacobi symbol. We also prove that $(10,9)_p=0$ for any prime $p\equiv5\pmod{12}$, and $[5,5]_p=0$ for any prime $p\equiv 13,17\pmod{20}$, which were also conjectured by Sun. Our proofs involve character sums over finite fields.
We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime $p\equiv 1\pmod 4$ and integer $a\not\equiv0\pmod p$, we prove that \begin{align*}&(-1)^{|\{1\le k<\frac p4:\ (\frac kp)=-1\}|}\prod_{1\le j \{ak^2\}_p\right\}\right| \\&+\left|\left\{(j,k):\ 1\le j \frac p2\right\}\right| \\\equiv&\left|\left\{1\le k<\frac p4:\ \left(\frac kp\right)=\left(\frac ap\right)\right\}\right|\pmod2. \end{align*} where $(\frac{a}p)$ is the Legendre symbol, $\varepsilon_p$ and $h(p)$ are the fundamental unit and the class number of the real quadratic field $\mathbb Q(\sqrt p)$ respectively, and $\{x\}_p$ is the least nonnegative residue of an integer $x$ modulo $p$. Also, for any prime $p\equiv3\pmod4$ and $δ=1,2$, we determine $$(-1)^{\left|\left\{(j,k): \ 1\le j \{δT_k\}_p\right\}\right|},$$ where $T_m$ denotes the triangular number $m(m+1)/2$.
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 $x+2y+3z=1$.
In this paper we investigate some new problems in additive combinatorics. Our problems mainly involve permutations (or circular permutations) $n$ distinct numbers (or elements of an additive abelian group) $a_1,\ldots,a_n$ with adjacent sums $a_i+a_{i+1}$ (or differences $a_i-a_{i+1}$) pairwise distinct. For an odd prime power $q=2n+1>13$ with $q\not=25$, we show that there is a circular permutation $(a_1,\ldots,a_n)$ of the elements of $S=\{a^2:\ a\in\mathbb F_q\setminus\{0\}\}$ such that $\{a_1+a_2,\ldots,a_{n-1}+a_n,a_n+a_1\}=S$, where $\mathbb F_q$ denotes the field of order $q$. For any finite subset $A$ of an additive torsion-free abelian group $G$ with $|A|=n>3$, we prove that there is a numbering $a_1,\ldots,a_n$ of the elements of $A$ such that $$a_1+2a_2,\ a_2+2a_3,\ \ldots,\ a_{n-1}+2a_n,\ a_n+2a_1$$ are pairwise distinct. We also pose 30 open conjectures for further research.
We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$, and $(\frac{\cdot}p)$ denotes the Legendre symbol. On the basis of our numerical computations, we formulate 35 conjectures involving primitive roots modulo primes. For example, we conjecture that for any prime $p$ there is a primitive root $g 3$ there is a prime $q 3$ there exists a Fibonacci number $F_k 3$.