Showing papers by "Zhi-Wei Sun published in 2016"

Two congruences involving harmonic numbers with applications

Abstract: The harmonic numbers Hn=∑0 3 be a prime. With the help of some combinatorial identities, we establish the following two new congruences: ∑k=1p−12k k k Hk ≡ 1 3 p 3 Bp−2 1 3 (mod p) and ∑k=1p−12k k k H2k ≡ 7 12 p 3 Bp−2 1 3 (mod p), where Bn(x) denotes the Bernoulli polynomial of degree n. As an application, we determine ∑n=1p−1g n and ∑n=1p−1h n modulo p3, where gn =∑k=0nn k22k k andhn =∑k=0nn k2C k with Ck = 2k k /(k + 1).

Refining Lagrange's four-square theorem

Abstract: Lagrange's four-square theorem asserts that any $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as the sum of four squares. This can be further refined in various ways. We show that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Z$ such that $x+y+z$ (or $x+2y$, $x+y+2z$) is a square (or a cube). We also prove that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb N$ such that $P(x,y,z)$ is a square, whenever $P(x,y,z)$ is among the polynomials \begin{gather*} x,\ 2x,\ x-y,\ 2x-2y,\ a(x^2-y^2)\ (a=1,2,3),\ x^2-3y^2,\ 3x^2-2y^2, \\x^2+ky^2\ (k=2,3,5,6,8,12),\ (x+4y+4z)^2+(9x+3y+3z)^2, \\x^2y^2+y^2z^2+z^2x^2,\ x^4+8y^3z+8yz^3, x^4+16y^3z+64yz^3. \end{gather*} We also pose some conjectures for further research; for example, our 1-3-5-Conjecture states that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb N$ such that $x+3y+5z$ is a square.

Congruences involving g_n(x)=\sum \limits _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k

Abstract: Define $$g_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k$$ for $$n=0,1,2,\ldots $$ . Those numbers $$g_n=g_n(1)$$ are closely related to Apery numbers and Franel numbers. In this paper we establish some fundamental congruences involving $$g_n(x)$$ . For example, for any prime $$p>5$$ we have $$\begin{aligned} \sum _{k=1}^{p-1}\frac{g_k(-1)}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{g_k(-1)}{k^2}\equiv 0\pmod p. \end{aligned}$$ This is similar to Wolstenholme’s classical congruences $$\begin{aligned} \sum _{k=1}^{p-1}\frac{1}{k}\equiv 0\pmod {p^2}\quad \text {and}\quad \sum _{k=1}^{p-1}\frac{1}{k^2}\equiv 0\pmod p \end{aligned}$$ for any prime $$p>3$$ .

Supercongruences involving Bernoulli polynomials

Abstract: Let p > 3 be a prime, and let a be a rational p-adic integer. Let {Bn} and {Bn(x)} denote the Bernoulli numbers and Bernoulli polynomials given by B0 = 1,∑k=0n−1n k Bk = 0 (n ≥ 2) and Bn(x) =∑k=0nn k Bkxn−k(n ≥ 0). In this paper, we show that ∑k=0p−1 a k −1 − a k 1 2k − 1 ≡−(2a + 1)(2t + 1) − p2t(t + 1)(4 + (2a + 1)B p−2(−a))(modp3), ∑k=0p−1 a k −1 − a k 2a + 1 2k + 1 ≡ 1 + 2t + p2t(t + 1)B p−2(−a)(modp3), where t = (a −〈a〉p)/p and 〈a〉p ∈{0, 1,…,p − 1} is given by a ≡〈a〉p(modp).

Some relations between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$

Abstract: Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$ $(x,y,z,w\in\Bbb Z$). In this paper we reveal some connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$.

On some universal sums of generalized polygonal numbers

Abstract: For m = 3; 4;::: those pm(x) = (m 2)x(x 1)=2 +x with x2 Z are called generalized m-gonal numbers. Sun (13) studied for what values of positive integers a;b;c the sum ap5 +bp5 +cp5 is universal over Z (i.e., any n2 N =f0; 1; 2;:::g has the form ap5(x) + bp5(y) + cp5(z) with x;y;z 2 Z). We prove that p5 + bp5 + 3p5 (b = 1; 2; 3; 4; 9) and p5 + 2p5 + 6p5 are universal over Z, as conjectured by Sun. Sun also conjectured that any n 2 N can be written as p3(x) + p5(y) + p11(z) and 3p3(x) + p5(y) + p7(z) with x;y;z 2 N; in contrast, we show that p3 + p5 + p11 and 3p3 + p5 + p7 are universal over Z. Our proofs are essentially elementary and hence suitable for general readers.

Telescoping method and congruences for double sums

Abstract: In recent years, Z.-W. Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double summations of hypergeometric terms. Using the telescoping method and certain mathematical software packages, we transform such a double summation into a single sum. With this new approach, we confirm several open conjectures of Sun.

On the number of representations of n as a linear combination of four triangular numbers II

Abstract: Let ℤ and ℕ be the set of integers and the set of positive integers, respectively. For a,b,c,d,n ∈ ℕ, let t(a,b,c,d; n) be the number of representations of n by ax(x − 1)/2 + by(y − 1)/2 + cz(z − 1)/2 + dw(w − 1)/2 (x,y,z,w∈ℤ). In this paper, we obtain explicit formulas for t(a,b,c,d; n) in the cases (a,b,c,d) = (1, 2, 2, 4), (1, 2, 4, 4), (1, 1, 4, 4), (1, 4, 4, 4), (1, 3, 9, 9), (1, 1, 3, 9), (1, 3, 3, 9), (1, 1, 9, 9), (1, 9, 9, 9) and (1, 1, 1, 9).

Arithmetic properties of Delannoy numbers and Schr\"oder numbers

Abstract: Define $$D_n(x)=\sum_{k=0}^n\binom nk^2x^k(x+1)^{n-k}\ \ \ \mbox{for}\ n=0,1,2,\ldots$$ and $$s_n(x)=\sum_{k=1}^n\frac1n\binom nk\binom n{k-1}x^{k-1}(x+1)^{n-k}\ \ \ \mbox{for}\ n=1,2,3,\ldots.$$ Then $D_n(1)$ is the $n$-th central Delannoy number $D_n$, and $s_n(1)$ is the $n$-th little Schroder number $s_n$. In this paper we obtain some surprising arithmetic properties of $D_n(x)$ and $s_n(x)$. We show that $$\frac1n\sum_{k=0}^{n-1}D_k(x)s_{k+1}(x)\in\mathbb Z[x(x+1)]\ \quad\mbox{for all}\ n=1,2,3,\ldots.$$ Moreover, for any odd prime $p$ and $p$-adic integer $x ot\equiv0,-1\pmod p$, we establish the supercongruence $$\sum_{k=0}^{p-1}D_k(x)s_{k+1}(x)\equiv0\pmod{p^2}.$$ As an application we confirm Conjecture 5.5 in [S14a], in particular we prove that $$\frac1n\sum_{k=0}^{n-1}T_kM_k(-3)^{n-1-k}\in\mathbb Z\quad\mbox{for all}\ n=1,2,3,\ldots,$$ where $T_k$ is the $k$-th central trinomial coefficient and $M_k$ is the $k$-th Motzkin number.

On a pair of zeta functions

Abstract: Let m be a positive integer, and define ζm(s) =∑n=1∞(−e2πi/m)ω(n) ns andζm∗(s) =∑ n=1∞(−e2πi/m)Ω(n) ns , for ℛ(s) > 1, where ω(n) denotes the number of distinct prime factors of n, and Ω(n) represents the total number of prime factors of n (counted with multiplicity). In this paper, we study these two zeta functions and related arithmetical functions. We show that ∑ n=1n is squarefree∞(−e2πi/m)ω(n) n = 0if m > 4, which is similar to the known identity ∑n=1∞μ(n)/n = 0 equivalent to the Prime Number Theorem. For m > 4, we prove that ζm(1) :=∑n=1∞(−e2πi/m)ω(n) n = 0andζm∗(1) :=∑ n=1∞(−e2πi/m)Ω(n) n = 0. We also raise a hypothesis on the parities of Ω(n) − n which implies the Riemann Hypothesis.

Hankel-type determinants for some combinatorial sequences

Abstract: 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 Apery 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.

A new result in combinatorial number theory

Abstract: Let $G$ be a finite abelian group with exponent $n>1$. For $a_1,\ldots,a_{n-1}\in G$, we determine completely when there is a permutation $\sigma$ on $\{1,\ldots,n-1\}$ such that $sa_{\sigma(s)} ot=0$ for all $s=1,\ldots,n-1$. When $G$ is the cyclic group $\mathbb Z/n\mathbb Z$, this confirms a conjecture of Z.-W. Sun.

A kind of orthogonal polynomials and related identities

Abstract: In this paper we introduce the polynomials $\{d_n^{(r)}(x)\}$ and $\{D_n^{(r)}(x)\}$ given by $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k} \ (n\ge 0)$, $D_0^{(r)}(x)=1,\ D_1^{(r)}(x)=x$ and $D_{n+1}^{(r)}(x)=xD_n^{(r)}(x)-n(n+2r)D_{n-1}^{(r)}(x)\ (n\ge 1).$ We show that $\{D_n^{(r)}(x)\}$ are orthogonal polynomials for $r>-\frac 12$, and establish many identities for $\{d_n^{(r)}(x)\}$ and $\{D_n^{(r)}(x)\}$, especially obtain a formula for $d_n^{(r)}(x)^2$ and the linearization formulas for $d_m^{(r)}(x)d_n^{(r)}(x)$ and $D_m^{(r)}(x)D_n^{(r)}(x)$. As an application we extend recent work of Sun and Guo.

Two Properties of Catalan-Larcombe-French Numbers.

Abstract: Let (P n) be the Catalan-Larcombe-French numbers. The numbers P n occur in the theory of elliptic integrals, and are related to the arithmetic-geometric-mean. In this paper we investigate the properties of the related sequence S n = P n /2 n instead, since S n is an Apéry-like sequence. We prove a congruence and an inequality for P n .

Sums of four polygonal numbers with coefficients

Abstract: Let $m\ge3$ be an integer. The polygonal numbers of order $m+2$ are given by $p_{m+2}(n)=m\binom n2+n$ $(n=0,1,2,\ldots)$. A famous claim of Fermat proved by Cauchy asserts that each nonnegative integer is the sum of $m+2$ polygonal numbers of order $m+2$. For $(a,b)=(1,1),(2,2),(1,3),(2,4)$, we study whether any sufficiently large integer can be expressed as $$p_{m+2}(x_1) + p_{m+2}(x_2) + ap_{m+2}(x_3) + bp_{m+2}(x_4)$$ with $x_1,x_2,x_3,x_4$ nonnegative integers. We show that the answer is positive if $(a,b)\in\{(1,3),(2,4)\}$, or $(a,b)=(1,1)\ \&\ 4\mid m$, or $(a,b)=(2,2)\ \&\ m ot\equiv 2\pmod 4$. In particular, we confirm a conjecture of Z.-W. Sun which states that any natural number can be written as $p_6(x_1) + p_6(x_2) + 2p_6(x_3) + 4p_6(x_4)$ with $x_1,x_2,x_3,x_4$ nonnegative integers.