arXiv subjects
Publications and source records attributed to Zhi-Wei Sun.
Whether the constant $$G=\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^2}=\frac1{1^2}-\frac1{3^2}+\frac1{5^2}-\frac1{7^2}+\cdots$$ introduced by Catalan in the nineteen century is irrational, is a long-standing open problem. In this paper we prove the irrationality of $G$ via using suitable weights.
Hilbert's Tenth Problem over the integers was solved negatively by Y. Matiyasevich in 1970. In this paper, we prove that there is no algorithm to determine for any polynomial equation $P(z_1,\ldots,z_{10})=0$ (with integer coefficients and ten unknowns) whether it has integer solutions. This improves the previous 11 unknowns theorem.
Via the circle method, we deduce that all sufficiently large positive integers are sums of two primes and six positive Fibonacci numbers, and are also sums of two primes and seven Lucas numbers. We also prove that each sufficiently large positive integer can be written as a sum of two primes, three Fibonacci numbers, and three Lucas numbers.
Let $K$ be any quadratic number field, and let $O_K$ be the ring of algebraic integers in $K$. In 1975 J. Denef proved that Hilbert's Tenth Problem over $O_K$ has a negative solution. In this paper we establish the following undecidability result: There is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{16})=0$ (with integer coefficients and 16 unknowns) has solutions over $O_K$. Moreover, when $K$ is a real quadratic field, we show that $15$ unknowns suffice for undecidability.
In 2016 J. Koenigsmann proved that $\mathbb Q\setminus\mathbb Z$ is diophantine over $\mathbb Q$, i.e., there is a polynomial $P(t,x_1,\ldots,x_{n})\in\mathbb Z[t,x_1,\ldots,x_{n}]$ such that for any rational number $t$ we have $$t\not\in\mathbb Z\iff \exists x_1,\ldots,x_{n}\in\mathbb Q\,[P(t,x_1,\ldots,x_{n})=0].$$ In this paper we show that we may take $n=7$ which improves the previous record $n=10$ obtained by Daans in 2024. (Actually we even extend this to any global field.) This, together with a previous result of Z.-W. Sun, implies that there is no algorithm to decide for any $F(x_1,\ldots,x_{16})\in\mathbb Z[x_1,\ldots,x_{16}]$ whether $$\forall x_1,\ldots,x_9\in\mathbb Q\exists y_1,\ldots,y_{7}\in\mathbb Q\,[F(x_1,\ldots,x_9,y_1,\ldots,y_{7})=0].$$
With generalized central binomial coefficients $ \binom{2x}{x}:=\frac{Γ(2x+1)}{[Γ(x+1)]^2}$ defined through Euler's gamma function, we represent deformed Apéry-like series \[ \mathscr A_{s,n}:=\sum_{k=1}^\infty\left.\!\frac{\partial^n}{\partial x^n}\frac{1}{x^s\binom{2x}{x}}\right|_{x=k} \] by multiple Clausen values (MCVs), which belong to a special class of cyclotomic multiple zeta values (CMZVs) at level $3$. For example, exploiting provable algebraic relations among MCVs, we show that \[\mathscr A_{1,5}=-\frac{9[495L(χ_{-3},6)-30π^{2}L(χ_{-3},4)-2π^{4}L(χ_{-3},2)]}{4}\]and\[\mathscr A_{4,4}=\frac{352ζ_{5,3}}{15}+\frac{752537π^{8}}{10206000},\]where $ L(χ_{-3},s):=\sum_{n=0}^\infty\left[(3n+1)^{-s}-(3n+2)^{-s}\right]$ and $ζ_{5,3}:=\sum_{m>n>0}m^{-5}n^{-3}$.
In this paper, we prove a conjecture of the second author by evaluating the determinant $$\det\left[x + \left(\frac{i-j}p\right) + \left(\frac ip\right)y + \left(\frac jp\right)z + \left(\frac{ij}p\right)w\right]_{0\le i,j\le(p-3)/2}$$ for any odd prime $p$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. In particular, the determinant is equal to $x$ when $p\equiv 3\pmod4$.
For integers $l>0$ and $m\geqslant0$, we introduce the numbers $$S_l^{(m)}(n) = \sum_{k_1,\ldots,k_l\in\mathbb N\atop k_1+\cdots+k_l = n} \binom n{k_1,\ldots,k_l}^m \ \ (n=0,1,2,\ldots),$$ and prove that for any prime $p$ not dividing $l+1$ we have the congruence $$\sum_{n=1}^{p-1}\frac{(-1)^{mn}}{n^{m-1}}S_l^{(m)}(n)\equiv0\pmod p.$$ We also obtain a $q$-analogue of this result. For the Domb numbers given by $$D(n)=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2(n-k)}{n-k}=S_4^{(2)}(n)\ \ (n=0,1,2,\ldots),$$ we confirm a previous conjecture which states that $$\sum_{n=1}^{p-1}\frac{D(n)}n\equiv\left(\frac p3\right)\frac 25pB_{p-2}\left(\frac13\right)\pmod{p^2}$$ for any prime $p$, where $(\frac p3)$ is the Legendre symbol, and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$.
In this paper, we investigate zero-sum problems of new types. For example, given $2n-1$ integers $a_1,\ldots,a_{2n-1}$ not divisible by an integer $n>1$, we prove that for some nonempty $I\subseteq\{1,\ldots,2n-1\}$ with $|I|\leqslant n$, the sum $\sum_{i\in I}a_i$ is divisible by $n$ but not divisible by $n^2$. We also pose several conjectures for further research.
Let $G$ be a finite group with $|G|=p^m$ where $p$ is a prime and $m$ is a positive integer. Let $k<p$. Let $a_1,\ldots,a_k\in G$ be pairwise distinct and let $b_1,\ldots,b_k\in G$. Then there exists a permutation $σ$ on $1,\ldots,k$ such that $a_1b_{σ(1)},\ldots,a_kb_{σ(k)}$ are pairwise distinct. This extends a theorem of Feng, Sun and Xiang, who proved that the conclusion holds in abelian $p$-groups.
In this paper we confirm several conjectures on determinants and permanents. For example, we prove that for any prime $p\equiv3\pmod 4$ the number $2\det[a_{jk}]_{0\le j,k\le (p-1)/2}$ is congruent to a square modulo $p$, where $a_{jk}=(\frac{j+k}{p})+(\frac{j^2+k^2}{p})$ with $(\frac{\cdot}{p})$ the Legendre symbol. We also prove that ${\rm per}[j^{k-1}]_{1\leq j,k\leq n-1}\equiv0\pmod n$ for any integer $n>1$ with $n\not\equiv2\pmod 4$.
The $q$-analogue of an integer $m$ is given by $[m]_q=(1-q^m)/(1-q)$. Let $a$ be an integer, and let $n$ be a positive odd integer. Via discrete Fourier transforms, we establish the following two identities: $$\det\left[\left[\left\lfloor\frac{aj-(a+1)k}n\right\rfloor\right]_q\right]_{1\leqslant j,k\leqslant n}=-\left(\frac{a(a+1)}n\right)q^{(1-3n)/2}$$ and $$\det\left[\left[\left\lceil\frac{(a+1)j-ak}n\right\rceil\right]_q\right]_{1\leqslant j,k\leqslant n}=\left(\frac{a(a+1)}n\right)q^{(n-1)/2},$$ where $(\frac{\cdot}n)$ denotes the Jacobi symbol.
Let $p$ be an odd prime, and define $$G_p(x)=\prod_{k=1}^{(p-1)/2}\left(x-e^{2πi k^2/p}\right).$$ In this paper we study values of $G_p(x)$ at roots of unity via Galois theory, and confirm some previous conjectures. For example, for any primitive tenth root $ζ$ of unity, we prove that $$G_p(ζ)=\begin{cases}(-1)^{|\{1\le k\le \frac {p+9}{10}:\ (\frac kp)=-1\}|} &\text{if}\ p\equiv21\pmod{40}, \\(-1)^{|\{1\le k\le\frac {p+1}{10}:\ (\frac kp)=-1\}|}ζ^{2}&\text{if}\ p\equiv 29\pmod{40}, \end{cases}$$ where $(\frac kp)$ denotes the Legendre symbol.
In this paper, we evaluate some series via the WZ method, and confirm several previous conjectures. For example, we prove the following two identities conjectured by the second author: $$\sum_{k=0}^{\infty} \frac{(28k^2 + 10k + 1) \binom{2k}{k}^5}{(6k + 1)(-64)^k \binom{3k}{k} \binom{6k}{3k}} = \frac{3}π$$ and $$\sum_{k=1}^\infty \frac{d^4}{dk^4}\left(\frac{(21k-8)Γ(k+1)^2}{k^3Γ(2k+1)}\right)=\frac{1959}2ζ(6)-432ζ(3)^2. $$
In this paper we collect over 150 new series identities (involving binomial coefficients) conjectured by the author in 2026. The values involved are related to $π$ or Riemann's zeta function or Dirichlet's $L$-function. For example, we conjecture that $$\sum_{k=0}^\infty\frac{16k+3}{(-202^2)^k}\binom{2k}kT_k(19,-20)T_{2k}(9,-5)=\frac{43\sqrt{101}}{75π},$$ where $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. The conjectures in this paper might interest some readers and stimulate further research.
For each $n=0,1,2,\ldots$, the central trinomial coefficient $T_n$ is the coefficient of $x^n$ in the expansion of $(x^2+x+1)^n$. Let $p>3$ be a prime, and let $n$ be any positive integer. In 2016, the second author conjectured that the quotient $(T_{pn}-T_n)/(pn)^2$ is always a $p$-adic integer. In this paper, we confirm this conjecture, and further prove that $$\frac{T_{pn}-T_n}{(pn)^2}\equiv\frac{T_{n-1}}6\left(\frac p3\right)B_{p-2}\left(\frac13\right)\pmod p,$$ where $(\frac p3)$ is the Legendre symbol and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$.
We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to reduce the corresponding series to special values of the Dirichlet $L$-functions. For example, we establish the following identities conjectured by Sun:\[\sum_{k=0}^\infty\binom{2k}{k}^3\left[ \mathsf H_{2k}^{(2)}-\frac{25}{92}\mathsf H_{ k}^{(2)} +\frac{735L_{-7}(2)-86π^{2}}{1104}\right]\frac{1}{4096^{k}}=0,\]\[\sum_{k=0}^\infty\binom{2k}k^3\left[\mathsf H_{2k}^{(3)}-\frac{43}{352}\mathsf H_k^{(3)}\right]\frac{42k+5}{4096^k}=\frac{555ζ(3)}{77π}-\frac{32G}{11},\] where $ \mathsf H^{(r)}_k:= \sum_{0<n\leq k}\frac{1}{n^r}$, $ L_{-7}(2):= \sum_{n=1}^\infty\left(\frac{-7}{n}\right)\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{4^{2}}-\frac{1}{5^{2}}-\frac{1}{6^{2}}+\frac{1}{8^{2}}+\cdots $, $ G:= \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^2}$, and $ ζ(3):= \sum_{n=1}^\infty\frac1{n^3}$.
An integer-valued polynomial $P(x,y,z)$ is said to be universal (over $\mathbb Z$) if each nonnegative integer can be written as $P(x,y,z)$ with $x,y,z\in\mathbb Z$. In this paper, we mainly introduce a new technique to determine the universality of some sums in the form $x(a_1x+a_2)/2+y(b_1y+b_2)/2+z(c_1z+c_2)/2$ (with $a_1-a_2,b_1-b_2,c_1-c_2$ all even) conjectured by Sun, using various identities of Ramanujan's theta functions. For example, we prove that $x(3x+1)+y(3y+2)+2z(3z+2)$ and $x(4x+r)+y(3y+1)/2+z(7z+1)/2\ (r=1,3)$ are universal.