arXiv subjects
Publications and source records attributed to Zhi-Hong Sun.
For a forbidden graph $L$, let $ex(p;L)$ denote the maximal number of edges in a simple graph of order $p$ not containing $L$. Let $T_n$ denote the unique tree on $n$ vertices with maximal degree $n-2$, and let $T_n^*=(V,E)$ be the tree on $n$ vertices with $V=\{v_0,v_1,\ldots,v_{n-1}\}$ and $E=\{v_0v_1,\ldots,v_0v_{n-3},v_{n-3}v_{n-2},v_{n-2}v_{n-1}\}$. In the paper we give exact values of $ex(p;T_n)$ and $ex(p;T_n^*)$.
For $n\ge 6$ let $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_1=\{v_0v_1,\ldots,v_0v_{n-4},v_1v_{n-3},v_1v_{n-2}$, $v_1v_{n-1}\}$, $E_2=\{v_0v_1,\ldots,v_0v_{n-4},v_1v_{n-3},v_1v_{n-2},v_2v_{n-1}\}$, $E_3=\{v_0v_1,\ldots,v_0v_{n-4}$, $v_1v_{n-3},v_2v_{n-2},v_3v_{n-1}\}$, $T_n^3=(V,E_1),\ T_n^{''}=(V,E_2)$ and $T_n^{'''} =(V,E_3).$ In this paper, for $p\ge n\ge 15$ we obtain explicit formulas for $ex(p;T_n^3)$, $ex(p;T_n^{''})$ and $ex(p;T_n^{'''})$, where $ex(p;L)$ denotes the maximal number of edges in a graph of order $p$ not containing $L$ as a subgraph.
Let $p>3$ be a prime, and let $a$ be a rational p-adic integer. Let $\{B_n(x)\}$ and $\{E_n(x)\}$ denote the Bernoulli polynomials and Euler polynomials, respectively. In this paper we show that $$\sum_{k=0}^{p-1}\binom ak\binom{-1-a}k\equiv (-1)^{\langle a\rangle_p}+ p^2t(t+1)E_{p-3}(-a)\pmod{p^3}$$ and for $a\not\equiv -\frac 12\pmod p$, $$\sum_{k=0}^{p-1}\binom ak\binom{-1-a}k\frac 1{2k+1}\equiv \frac{1+2t}{1+2a} +p^2\frac{t(t+1)}{1+2a}B_{p-2}(-a)\pmod{p^3},$$ where $\langle a\rangle_p\in\{0,1,\ldots,p-1\}$ satisfying $a\equiv \langle a\rangle_p\pmod p$ and $t=(a-\langle a\rangle_p)/p$. Taking $a=-\frac 13,-\frac 14,-\frac 16$ in the above congruences we solve some conjectures of Z.W. Sun. In this paper we also establish congruences for $\sum_{k=0}^{p-1}k\binom ak\binom{-1-a}k,\ \sum_{k=0}^{p-1}\binom ak\binom{-1-a}k\frac 1{2k-1},\ \sum_{k=1}^{p-1}\frac 1k\binom ak\binom{-1-a}k\pmod{p^3}$ and $\sum_{k=1}^{p-1}\frac {(-1)^k}k\binom ak,\ \sum_{k=0}^{p-1}\binom ak(-2)^k\pmod{p^2}.$
Let $\{B_n\}$, $\{B_n(x)\}$ and $\{E_n(x)\}$ be the Bernoulli numbers, Bernoulli polynomials and Euler polynomials, respectively. In this paper we mainly establish formulas for $\sum_{6\mid k-3}\binom nkB_{n-k}(x)$, $\sum_{6\mid k}\binom nkE_{n-k}(x)$ and $\sum_{6\mid k-3}\binom nkm^kB_{n-k}$ in the cases $m=2,3,4$.
In this paper we continue to investigate the properties of those sequences $\{a_n\}$ satisfying the condition $\sum_{k=0}^n\binom nk(-1)^ka_k=\pm a_n$ $(n\ge 0)$. As applications we deduce new recurrence relations and congruences for Bernoulli and Euler numbers.
Let $\Bbb Z$ be the set of integers, and let $p$ be a prime of the form $4k+1$. Suppose $q\in\Bbb Z$, $2\nmid q$, $p\nmid q$, $p=c^2+d^2$, $c,d\in\Bbb Z$ and $c\equiv 1\pmod 4$. In this paper we continue to discuss congruences for $q^{[p/8]}\pmod p$ and present new reciprocity laws, but we assume $4p=x^2+qy^2$ or $p=x^2+2qy^2$, where $[\cdot]$ is the greatest integer function and $x,y\in\Bbb Z$.
Let $P,Q\in\Bbb Z$, $U_0=0,\ U_1=1$ and $U_{n+1}=PU_n-QU_{n+1}$. In this paper we obtain a general congruence for $U_{kmn^r}/U_k\pmod {n^{r+1}}$, where $k,m,n,r$ are positive integers. As applications we extend Lucas' law of repetition and characterize the square prime factors of $a^n+1$ or $S_n$, where $\{S_n\}$ is given by $S_1=P^2+2$ and $S_{k+1}=S_k^2-2\ (k\ge 1)$.
Let $p$ be an odd prime and let $m\not\equiv 0\pmod p$ be a rational p-adic integer. In this paper we reveal the connection between quartic residues and the sum $\sum_{k=0}^{[p/4]}\binom{4k}{2k}\frac 1{m^k}$, where $[x]$ is the greatest integer not exceeding $x$. Let $q$ be a prime of the form $4k+1$ and so $q=a^2+b^2$ with $a,b\in\Bbb Z$. When $p\nmid ab(a^2-b^2)q$, we show that for $r=0,1,2,3$, $p^{\frac{q-1}4}\equiv (\frac ab)^r\pmod q$ if and only if $$\sum_{k=0}^{[p/4]}\binom{4k}{2k}\Big(\frac{a^2}{16q}\Big)^k\equiv (-1)^{\frac{p^2-1}8a+\frac{p-1}2\cdot \frac{q-1}4}\Big(\frac pq\Big) \Big(\frac ab\Big)^r\pmod p,$$ where $(\frac pq)$ is the Legendre symbol. We also establish congruences for $\sum_{k=0}^{[p/4]}\binom{4k}{2k}\frac 1{m^k}\pmod p$ in the cases $m=17,18,20,32,52,80,272$.
Let $p$ be a prime greater than $3$ and let $a$ be a rational p-adic integer. In this paper we try to determine $\sum_{k=1}^{[p/3]}\binom{3k}ka^k\pmod p$, and real the connection between cubic congruences and the sum $\sum_{k=1}^{[p/3]}\binom{3k}ka^k$, where $[x]$ is the greatest integer not exceeding $x$. Suppose that $a_1,a_2,a_3$ are rational p-adic integers, $P=-2a_1^3+9a_1a_2-27a_3$, $Q=(a_1^2-3a_2)^3$ and $PQ(P^2-Q)(P^2-3Q)(P^2-4Q)\not\equiv 0\pmod p$. In this paper we show that the number of solutions of the congruence $x^3+a_1x^2+a_2x+a_3\equiv 0\pmod p$ depends only on $\sum_{k=1}^{[p/3]}\binom{3k}k(\frac{4Q-P^2}{27Q})^k\pmod p$. Let $q$ be a prime of the form $3k+1$ and so $4q=L^2+27M^2$ with $L,M\in\Bbb Z$. When $p\not=q$ and $p\nmid L$, we establish congruences for $\sum_{k=1}^{[p/3]}\binom{3k}k(\frac{M^2}q)^k$ and $\sum_{k=1}^{[p/3]}\binom{3k}k(\frac{L^2}{27q})^k$ modulo p. As a consequence, we show that $x^3-qx-qM\equiv 0\pmod p$ has three solutions if and only if $p$ is a cubic residue of $q$.
In this paper we prove several inequalities for binomial coefficients. For instance, if $ k$ and $n$ are positive integers such that $n\ge 400$ and $[\frac n5]\le k\le [\frac n2]$, where $[x]$ is the greatest integer not exceeding $x$, then $$\binom nk<\Big(1-\frac{5(k-[\f n5])}{6n^2}\Big) \frac{n^{n-\f 12}}{k^k(n-k)^{n-k}}.$$
For $n\ge 3$ let $f(n)$ be the least positive integer $k$ such that $\binom nk>\frac{2^n}{n+1}$. In this paper we investigate the properties of $f(n)$.
For a given real number $a$ we define the sequence $\{E_{n,a}\}$ by $E_{0,a}=1$ and $E_{n,a}=-a\sum_{k=1}^{[n/2]} \binom n{2k}E_{n-2k,a}$ $(n\ge 1)$, where $[x]$ is the greatest integer not exceeding $x$. Since $E_{n,1}=E_n$ is the n-th Euler number, $E_{n,a}$ can be viewed as a natural generalization of Euler numbers. In this paper we deduce some identities and an inversion formula involving $\{E_{n,a}\}$, and establish congruences for $E_{2n,a}\mod{2^{{\rm ord}_2n+8}}$, $E_{2n,a}\pmod{3^{{\rm ord}_3n+5}}$ and $E_{2n,a}\pmod{5^{{\rm ord}_5n+4}}$ provided that $a$ is a nonzero integer, where ${\rm ord}_pn$ is the least nonnegative integer $α$ such that $p^{\a}\mid n$ but $p^{\a+1}\nmid n$.
For a nonzero integer $a$ let ${E_n^{(a)}}$ be given by $\sum_{k=0}^{[n/2]}\binom n{2k}a^{2k}E_{n-2k}^{(a)}=(1-a)^n$ $(n=0,1,2,...)$, where $[x]$ is the greatest integer not exceeding $x$. As $E_n^{(1)}=E_n$ is the Euler number, $E_n^{(a)}$ can be viewed as a generalization of Euler numbers. Let $k$ and $m$ be positive integers, and let $b$ be a nonnegative integer. In this paper, we determine $E_{2^mk+b}^{(a)}$ modulo $ 2^{m+10}$ for $m\ge 5$. For $m\ge 5$ we also establish congruences for $U_{kφ{(5^m)}+b},\; E_{kφ{(5^m)}+b},\; S_{kφ{(5^m)}+b}\pmod{5^{m+5}}$ and $S_{kφ{(3^m)}+b}\pmod{3^{m+5}},$ where $U_{2n}=E_{2n}^{(3/2)}$, $S_n=E_n^{(2)}$ and $φ(n)$ is Euler's function.
Let $p$ be an odd prime. In the paper we collect the author's various conjectures on congruences modulo $p$ or $p^2$, which are concerned with sums of binomial coefficients, Lucas sequences, power residues and special binary quadratic forms.
Let $p>3$ be a prime, and let $R_p$ be the set of rational numbers whose denominator is coprime to $p$. Let $\{P_n(x)\}$ be the Legendre polynomials. In this paper we mainly show that for $m,n,t\in R_p$ with $m\not\e 0\pmod p$, $$\align &P_{[\frac p6]}(t) \e -\Big(\frac 3p\Big)\sum_{x=0}^{p-1}\Big(\frac{x^3-3x+2t}p\Big)\pmod p, &\Big(\sum_{x=0}^{p-1}\Big(\frac{x^3+mx+n}p\Big)\Big)^2\equiv \Big(\frac{-3m}p\Big) \sum_{k=0}^{[p/6]}\binom{2k}k\binom{3k}k\binom{6k}{3k} \Big(\frac{4m^3+27n^2}{12^3\cdot 4m^3}\Big)^k\pmod p,$$ where $(\frac ap)$ is the Legendre symbol and $[x]$ is the greatest integer function. As an application we solve some conjectures of Z.W. Sun and the author concerning $\sum_{k=0}^{p-1}\binom{2k}k\binom{3k}k\binom{6k}{3k}/m^k\pmod {p^2}$, where $m$ is an integer not divisible by $p$.
Let $\Bbb Z$ be the set of integers, and let $(m,n)$ be the greatest common divisor of integers $m$ and $n$. Let $p\equiv 1\mod 4$ be a prime, $q\in\Bbb Z$, $2\nmid q$ and $p=c^2+d^2=x^2+qy^2$ with $c,d,x,y\in\Bbb Z$ and $c\e 1\mod 4$. Suppose that $(c,x+d)=1$ or $(d,x+c)$ is a power of 2. In the paper, by using the quartic reciprocity law we determine $q^{[p/8]}\mod p$ in terms of $c,d,x$ and $y$, where $[\cdot]$ is the greatest integer function. We also determine $\big(\frac{b+\sqrt{b^2+4^α}}2\big)^{\frac{p-1}4}\mod p$ for odd $b$ and $(2a+\sqrt{4a^2+1})^{\f{p-1}4}\mod p$ for $a\in\Bbb Z$. As applications we obtain the congruence for $U_{\f{p-1}4}\mod p$ and the criterion for $p\mid U_{\frac{p-1}8}$ (if $p\equiv 1\mod 8$), where $\{U_n\}$ is the Lucas sequence given by $U_0=0,\ U_1=1$ and $U_{n+1}=bU_n+U_{n-1}\ (n\ge 1)$, and $b\not\equiv 2\mod 4$. Hence we partially solve some conjectures posed by the author in two previous papers.
Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper we solve some conjectures of Z.W. Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^3/m^k\pmod{p^2}$, $\sum_{k=0}^{p-1}\binom{2k}k\b{4k}{2k}/m^k\pmod p$ and $\sum_{k=0}^{p-1}\binom{2k}k^2\b{4k}{2k}/m^k\pmod {p^2}.$ In particular, we show that $\sum_{k=0}^{\frac{p-1}{2}}\binom{2k}k^3\equiv 0\pmod {p^2}$ for $p\equiv 3,5,6\pmod 7$. Let $P_n(x)$ be the Legendre polynomials. In the paper we also show that $ P_{[\frac {p}{4}]}(t)\equiv -\big(\frac{-6}{p}\big)\sum_{x=0}^{p-1} \big(\frac{x^3-3/2(3t+5)x-9t-7}{p}\big)\pmod p$ and determine $P_{\frac{p-1}{2}}(\sqrt 2), P_{\frac{p-1}{2}}(\frac{3\sqrt 2}{4}), P_{\frac{p-1}{2}}(\sqrt {-3}),P_{\frac{p-1}{2}}(\frac{\sqrt 3}{2}), P_{\frac{p-1}{2}}(\sqrt {-63}), P_{\frac{p-1}{2}}(\frac {3\sqrt 7}{8}) \pmod p$, where $t$ is a rational $p-$integer, $[x]$ is the greatest integer not exceeding $x$ and $(\frac {a}{p})$ is the Legendre symbol. As consequences we determine $P_{[\frac {p}{4}]}(t)\pmod p$ in the cases $t=-5/3,-7/9,-65/63$ and confirm many conjectures of Z.W. Sun.
Let $\{E_n\}$ be the Euler numbers. In the paper we determine $E_{2^mk+b}-E_b$ modulo $2^{m+7}$, where $k$ and $m$ are positive integers and $b\in{0,2,4,...}$.