arXiv subjects
Publications and source records attributed to Zhi-Wei Sun.
Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. In this paper, we study the determinant $$\det\left[\left(\frac{j^2-k^2}p\right)+\left(\frac{jk}p\right)w\right]_{δ\le j,k\le (p-1)/2}$$ with $δ\in\{0,1\}$. For example, we prove that the determinant does not depend on $w$ if $p\equiv3\pmod4$ and $δ=0$.
In this paper we investigate determinants whose entries are linear combinations of Legendre symbols. We deduce some new results in this direction; for example, we prove that for any prime $p\equiv3\pmod4$ we have $$\det\left[x+\left(\frac{j-k}p\right)+\left(\frac jp\right)-\left(\frac kp\right)\right]_{0\le j,k\le(p-1)/2}=4,$$ where $(\frac{\cdot}p)$ is the Legendre symbol. We also pose many conjectures for further research. For example, for any prime $p>3$ we conjecture that \begin{align*}&\ \det\left[\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)+\left(\frac{jk}p\right)\right]_{1\le j,k\le(p-1)/2} \\=&\ \begin{cases}(\frac 2p)p^{(p-5)/4}&\text{if}\ p\equiv1\pmod4, \\(-1)^{(h(-p)-1)/2}(1-(2-(\frac 2p))h(-p))p^{(p-3)/4}&\text{if}\ p\equiv3\pmod4, \end{cases}\end{align*} where $h(-p)$ is the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$.
In this paper we evaluate several determinants involving quadratic residues modulo primes. For example, for any prime $p>3$ with $p\equiv3\pmod4$ and $a,b\in\mathbb Z$ with $p\nmid ab$, we prove that $$\det\left[ 1+\tanπ\frac{aj^2+bk^2}p \right]_{1\le j,k \le \frac{p-1}2} = \begin{cases}-2^{(p-1)/2}p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=1, \\p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=-1, \end{cases}$$ where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also pose some conjectures for further research.
In this paper we first review the history of Hilbert's Tenth Problem, and then study mixed quantifier prefixes over Diophantine equations with integer variables. For example, we prove that $\forall^2\exists^4$ over $\mathbb Z$ is undecidable, that is, there is no algorithm to determine for any $P(x_1,\ldots,x_6)\in\mathbb Z[x_1,\ldots,x_6]$ whether $$\forall x_1\forall x_2\exists x_3\exists x_4\exists x_5\exists x_6(P(x_1,\ldots,x_6)=0),$$ where $x_1,\ldots,x_6$ are integer variables. We also have some similar undecidable results with universal quantifies bounded, for example, $\exists^2\forall^2\exists^2$ over $\mathbb Z$ with $\forall$ bounded is undecidable. We conjecture that $\forall^2\exists^2$ over $\mathbb Z$ is undecidable.
Let $p>3$ be a prime, and let $d\in\mathbb Z$ with $p\nmid d$. For the determinants $$S_m(d,p)=\det\left[(i^2+dj^2)^{m}\right]_{1\leqslant i,j \leqslant (p-1)/2}\ \ \left(\frac{p-1}2\leqslant m\leqslant p-1\right),$$ Sun recently determined $S_m(d,p)$ modulo $p$ when $m\in\{p-2,p-3\}$ and $(\frac {-d}p)=-1$. In this paper, we obtain $S_{p-2}(d,p)$ modulo $p$ in the remaining case $(\frac{-d}p)=1$, and determine the Legendre symbols $(\frac{S_{p-3}\,(d,p)}p)$ and $(\frac{S_{p-4}\,(d,p)}p)$ in some special cases.
In this paper we study some determinants and permanents. In particular, we investigate the new type determinants $$\det[(i^2+cij+dj^2)^{p-2}]_{1\le i,j\le p-1}\ \text{and} \ \det[(i^2+cij+dj^2)^{p-2}]_{0\le i,j\le p-1}$$ modulo an odd prime $p$, where $c$ and $d$ are integers. We also pose some conjectures for further research.
Using cyclotomic multiple zeta values of level $8$, we confirm and generalize several conjectural identities on infinite series with summands involving $\binom{2k}k8^k/(\binom{3k}k\binom{6k}{3k})$. For example, we prove that \[\sum_{k=0}^\infty\frac{(350k-17)\binom{2k}k8^k} {\binom{3k}k\binom{6k}{3k}}=15\sqrt2\,π+27\] and \[\sum_{k=1}^\infty\frac{\left\{(5k-1)\left[16\mathsf H_{2k-1}^{(2)}-3\mathsf H_{k-1}^{(2)}\right]-\frac{12(6k-1)}{(2k-1)^2}\right\}\binom{2k}k8^k} {k(2k-1)\binom{3k}k\binom{6k}{3k}}=\frac{π^3}{12\sqrt2},\] where $\mathsf H^{(2)}_m$ denotes the second-order harmonic number $\sum_{0<j\leq m}\frac1{j^2}$.
We perform polylogarithmic reductions for several classes of infinite sums motivated by Z.-W. Sun's related works in 2022--2023. For certain choices of parameters, these series can be expressed by cyclotomic multiple zeta values of levels $4$, $5$, $6$, $7$, $8$, $9$, $10$, and $12$. In particular, we obtain closed forms of the series $$\sum_{k=0}^\infty\frac{x_0^k}{(k+1)\binom{3k}k} \ \ \text{and}\ \ \sum_{k=1}^\infty\frac{x_0^k}{k^2\binom{3k}k}$$ for any $x_0\in(-27/4,27/4)$.
Let $d>r\ge 0$ be integers. For positive integers $a,b,c$, if any term of the arithmetic progression $\{r+dn:\ n=0,1,2,\ldots\}$ can be written as $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb{Z}$, then the form $ax^2+by^2+cz^2$ is called $(d,r)$-universal. In this paper, via the theory of ternary quadratic forms we study the $(d,r)$-universality of some diagonal ternary quadratic forms conjectured by L. Pehlivan and K. S. Williams, and Z.-W. Sun. For example, we prove that $2x^2+3y^2+10z^2$ is $(8,5)$-universal, $x^2+3y^2+8z^2$ and $x^2+2y^2+12z^2$ are $(10,1)$-universal and $(10,9)$-universal, and $3x^2+5y^2+15z^2$ is $(15,8)$-universal.
Let $p>3$ be a prime. For any $p$-adic integer $a$, we determine $$\sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}kH_k,\ \ \sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}kH_k^{(2)},\ \ \sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}k\frac{H_k^{(2)}}{2k+1}$$ modulo $p^2$, where $H_k=\sum_{0<j\le k}1/j$ and $H_k^{(2)}=\sum_{0<j\le k}1/j^2$. In particular, we show that \begin{gather*}\sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}kH_k\equiv(-1)^{\langle a\rangle_p}\,2\left(B_{p-1}(a)-B_{p-1}\right)\pmod p, \\\sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}kH_k^{(2)}\equiv -E_{p-3}(a)\pmod p, \\(2a-1)\sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}k\frac{H_k^{(2)}}{2k+1}\equiv B_{p-2}(a)\pmod p, \end{gather*} where $\langle a\rangle_p$ stands for the least nonnegative integer $r$ with $a\equiv r\pmod{p}$, and $B_n(x)$ and $E_n(x)$ denote the Bernoulli polynomial of degree $n$ and the Euler polynomial of degree $n$ respectively. We also pose some new conjectures on congruences.
Let $p$ be an odd prime and let $a,b\in\mathbb Z$ with $p\nmid ab$. In this paper we mainly evaluate $$T_p^{(δ)}(a,b,x):=\det\left[x+\tanπ\frac{aj^2+bk^2}p\right]_{δ\le j,k\le (p-1)/2}\ \ (δ=0,1).$$ For example, in the case $p\equiv3\pmod4$ we show that $T_p^{(1)}(a,b,0)=0$ and $$T_p^{(0)}(a,b,x)=\begin{cases} 2^{(p-1)/2}p^{(p+1)/4}&\text{if}\ (\frac{ab}p)=1, \\p^{(p+1)/4}&\text{if}\ (\frac{ab}p)=-1,\end{cases}$$ where $(\frac{\cdot}p)$ is the Legendre symbol. When $(\frac{-ab}p)=-1$, we also evaluate the determinant $\det[x+\cotπ\frac{aj^2+bk^2}p]_{1\le j,k\le(p-1)/2}.$ In addition, we pose several conjectures one of which states that for any prime $p\equiv3\pmod4$ there is an integer $x_p\equiv1\pmod p$ such that $$\det\left[\sec2π\frac{(j-k)^2}p\right]_{0\le j,k\le p-1}=-p^{(p+3)/2}x_p^2.$$
Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv1\pmod{2m}$ and $2$ is an $m$th power residue modulo $p$, we determine the value of the product $\prod_{k\in R_m(p)}\tanπ\frac{ak}p$, where $$R_m(p)=\{0<k<p:\ k\in\mathbb Z\ \text{is an}\ m\text{th power reside modulo}\ p\}.$$ In particular, if $p=x^2+64y^2$ with $x,y\in\mathbb Z$, then $$\prod_{k\in R_4(p)}\left(1+\tanπ\frac {ak}p\right)=(-1)^{y}(-2)^{(p-1)/8}.$$
Let $p$ be an odd prime. For $b,c\in\mathbb Z$, Sun introduced the determinant $$D_p(b,c)=\left|(i^2+bij+cj^2)^{p-2}\right|_{1\leqslant i,j \leqslant p-1},$$ and investigated the Legendre symbol $(\frac{D_p(b,c)}p)$. Recently Wu, She and Ni proved that $(\frac{D_p(1,1)}p)=(\frac {-2}p)$ if $p\equiv2\pmod 3$, which confirms a previous conjecture of Sun. In this paper we determine $(\frac{D_p(1,1)}p)$ in the case $p\equiv1\pmod3$. Sun proved that $D_p(2,2)\equiv0\pmod p$ if $p\equiv3\pmod4$, in contrast we prove that $(\frac{D_p(2,2)}p)=1$ if $p\equiv1\pmod8$, and $(\frac{D_p(2,2)}p)=0$ if $p\equiv5\pmod8$. Our tools include generalized trinomial coefficients and Lucas sequences.
In this paper, we mainly establish a congruence for a sum involving Apéry numbers, which was conjectured by Z.-W. Sun. Namely, for any prime $p>3$ and positive odd integer $m$, we prove that there is a $p$-adic integer $c_m$ only depending on $m$ such that $$\sum_{k=0}^{p-1}(2k+1)^{m}(-1)^kA_k\equiv c_mp\left(\frac{p}{3}\right)\pmod{p^3},$$ where $A_k=\sum_{j=0}^{k}\binom{k}{j}^2\binom{k+j}{j}^2$ is the Apéry number and $(\frac{\cdot}{p})$ is the Legendre symbol.
Let $n$ be a positive integer, and let $A$ be a set of $k\ge 2n-1$ integers. For the restricted sumset $$ S_n(A)=\{a_1+\cdots +a_n:\ a_1,\ldots,a_n\in A,\ \text{and}\ a_i^2\neq a_j^2\ \text{for} \ 1\le i<j\le n\}, $$ by a 2002 result of Liu and Sun we have $$|S_n(A)|\ge (k-1)n-\frac 32n(n-1)+1.$$ In this paper, we determine the structure of $A$ when the lower bound is attained.
In 2016 J. Koenigsmann refined a celebrated theorem of J. Robinson by proving 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 $$t\not\in\mathbb Z\iff \exists x_1\cdots\exists x_{n}[P(t,x_1,\ldots,x_{n})=0]$$ where variables range over $\mathbb Q$, equivalently $$t\in\mathbb Z\iff \forall x_1\cdots\forall x_{n}[P(t,x_1,\ldots,x_{n})\not=0].$$ In this paper we prove that we may take $n=32$. Combining this with a result of Z.-W. Sun, we show that there is no algorithm to decide for any $f(x_1,\ldots,x_{41})\in\mathbb Z[x_1,\ldots,x_{41}]$ whether $$\forall x_1\cdots\forall x_9\exists y_1\cdots\exists y_{32}[f(x_1,\ldots,x_9,y_1,\ldots,y_{32})=0],$$ where variables range over $\mathbb Q$.
Let $n>1$ be an integer, and let $T$ be a tree with $n+1$ vertices $v_1,\ldots,v_{n+1}$, where $v_1$ and $v_{n+1}$ are two leaves of $T$. For each edge $e$ of $T$, assign a complex number $w(e)$ as its weight. We obtain that $$\det[x+d(v_{j+1},v_k)]_{1\le j,k\le n}=2^{n-2}\prod_{e\in E(T)}w(e),$$ where $d(v_{j+1},v_k)$ is the weighted distance between $v_{j+1}$ and $v_k$ in the tree $T$. This is similar to the celebrated Graham-Pollak theorem on determinants of distance matrices for trees. Actually, a more general result is deduced in this paper.
For any $m,n\in\mathbb{N}=\{0,1,2\ldots\}$, the truncated hypergeometric series ${}_{m+1}F_m$ is defined by $$ {}_{m+1}F_m\bigg[\begin{matrix}x_0&x_1&\ldots&x_m\\ &y_1&\ldots&y_m\end{matrix}\bigg|z\bigg]_n=\sum_{k=0}^n\frac{(x_0)_k(x_1)_k\cdots(x_m)_k}{(y_1)_k\cdots(y_m)_k}\cdot\frac{z^k}{k!}, $$ where $(x)_k=x(x+1)\cdots(x+k-1)$ is the Pochhammer symbol. Let $p$ be an odd prime. For $α,z\in\mathbb{Z}_p$ with $\langle -α\rangle_p\equiv0\pmod{2}$, where $\langle x\rangle_p$ denotes the least nonnegative residue of $x$ modulo $p$ for any $x\in\mathbb{Z}_p$, we mainly prove the following congruence motivated by Orr's identity: $$ {}_2F_1\bigg[\begin{matrix}\frac12α&\frac32-\frac12α\\ &1\end{matrix}\bigg|z\bigg]_{p-1}{}_2F_1\bigg[\begin{matrix}\frac12α&\frac12-\frac12α\\ &1\end{matrix}\bigg|z\bigg]_{p-1}\equiv{}_3F_2\bigg[\begin{matrix}α&2-α&\frac12\\ &1&1\end{matrix}\bigg|z\bigg]_{p-1}\pmod{p^2}. $$ As a corollary, for any positive integer $b$ with $p\equiv\pm1\pmod{b}$ and $\langle -1/b\rangle_p\equiv0\pmod{2}$, we deduce that $$ \sum_{k=0}^{p-1}(b^2k+b-1)\frac{\binom{2k}{k}}{4^k}\binom{-1/b}{k}\binom{1/b-1}{k}\equiv0\pmod{p^2}. $$ This confirms a conjectural congruence of the second author.