An extension of a curious binomial identity
In 2002 Zhi-Wei Sun [Integers 2(2002)] published a curious identity involving binomial coefficients. In this paper we present a generalization of the identity.
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arXiv subjects
Zhi-Wei Sun
Publications and source records attributed to Zhi-Wei Sun.
Congruences for sums of binomial coefficients
Let q>1 and m>0 be relatively prime integers. We find an explicit period $ν_m(q)$ such that for any integers n>0 and r we have $[n+ν_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q), or a=1 (mod q) and 2|m, where $[n,r]_m(a)=\sum_{k=r(mod m)}\binom{n}{k}a^k$. This is a further extension of a congruence of Glaisher.
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Combinatorial congruences modulo prime powers
Let p be any prime, and let a and n be nonnegative integers. Let $r\in Z$ and $f(x)\in Z[x]$. We establish the congruence $$p^{°f}\sum_{k=r(mod p^a)}\binom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{\sum_{i=a}^{\infty}[n/p^i]})$$ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas' theorem: If a is greater than one, and $l,s,t$ are nonnegative integers with $s,t<p$, then $$\frac{1}{[n/p^{a-1}]!} \sum_{k=r(mod p^a)} \binom{pn+s}{pk+t}(-1)^{pk}((k-r)/p^{a-1})^l =\frac {1}{[n/p^{a-1}]!} \sum_{k=r(mod p^a)}\binom{n}{k}\binom{s}{t}(-1)^k((k-r)/p^{a-1})^l (mod p).$$ We also present an application of the first congruence to Bernoulli polynomials, and apply the second congruence to show that a p-adic order bound given by the authors in a previous paper can be attained when p=2.
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On Fleck quotients
Let $p$ be a prime, and let $n>0$ and $r$ be integers. In this paper we study Fleck's quotient $$F_p(n,r)=(-p)^{-\lfloor(n-1)/(p-1)\rfloor} \sum_{k=r(mod p)}\binom {n}{k}(-1)^k\in Z.$$ We determine $F_p(n,r)$ mod $p$ completely by certain number-theoretic and combinatorial methods; consequently, if $2\le n\le p$ then $$\sum_{k=1}^n(-1)^{pk-1}\binom{pn-1}{pk-1} \equiv(n-1)!B_{p-n}p^n (mod p^{n+1}),$$ where $B_0,B_1,...$ are Bernoulli numbers. We also establish the Kummer-type congruence $F_p(n+p^a(p-1),r)\equiv F_p(n,r) (mod p^a)$ for $a=1,2,3,...$, and reveal some connections between Fleck's quotients and class numbers of the quadratic fields $\Q(\sqrt{\pm p})$ and the $p$-th cyclotomic field $\Q(ζ_p)$. In addition, generalized Fleck quotients are also studied in this paper.
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Mixed sums of squares and triangular numbers
By means of $q$-series, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus $x^2+y^2$ for some integers $x$ and $y$ with $x\not\equiv y (mod 2)$ or $x=y>0$. The paper also contains some other results and open conjectures on mixed sums of squares and triangular numbers.
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Combinatorial congruences and Stirling numbers
In this paper we obtain some sophisticated combinatorial congruences involving binomial coefficients and confirm two conjectures of the author and Davis. They are closely related to our investigation of the periodicity of the sequence $\sum_{j=0}^l{l\choose j}S(j,m)a^{l-j}(l=m,m+1,...)$ modulo a prime $p$, where $a$ and $m>0$ are integers, and those $S(j,m)$ are Stirling numbers of the second kind. We also give a new extension of Glaisher's congruence by showing that $(p-1)p^{[\log_p m]}$ is a period of the sequence $\sum_{j=r(mod p-1)}{l\choose j}S(j,m)(l=m,m+1,...)$ modulo $p$.
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Simple arguments on consecutive power residues
By some extremely simple arguments, we point out the following: (i) If n is the least positive k-th power non-residue modulo a positive integer m, then the greatest number of consecutive k-th power residues mod m is smaller than m/n. (ii) Let O_K be the ring of algebraic integers in a quadratic field $K=Q(\sqrt d)$ with d in {-1,-2,-3,-7,-11}. Then, for any irreducible $π\in O_K$ and positive integer k not relatively prime to $π\barπ-1$, there exists a k-th power non-residue $ω\in O_K$ modulo $π$ such that $|ω|<\sqrt{|π|}+0.65$.
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On the unique representability of spikes over prime fields
For an integer $n>2$, a rank-$n$ matroid is called an $n$-spike if it consists of $n$ three-point lines through a common point such that, for all $k\in\{1, 2, ..., n - 1\}$, the union of every set of $k$ of these lines has rank $k+1$. Spikes are very special and important in matroid theory. In 2003 Wu found the exact numbers of $n$-spikes over fields with 2, 3, 4, 5, 7 elements, and the asymptotic values for larger finite fields. In this paper, we prove that, for each prime number $p$, a $GF(p$) representable $n$-spike $M$ is only representable on fields with characteristic $p$ provided that $n \ge 2p-1$. Moreover, $M$ is uniquely representable over $GF(p)$.
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A number-theoretic approach to homotopy exponents of SU(n)
We use methods of combinatorial number theory to prove that, for each $n>1$ and any prime $p$, some homotopy group $π_i(SU(n))$ contains an element of order $p^{n-1+ord_p([n/p]!)}$, where $ord_p(m)$ denotes the largest integer $α$ such that $p^α$ divides $m$.
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Identities concerning Bernoulli and Euler polynomials
We establish two general identities for Bernoulli and Euler polynomials, which are of a new type and have many consequences. The most striking result in this paper is as follows: If $n$ is a positive integer, $r+s+t=n$ and $x+y+z=1$, then we have $$rF(s,t;x,y)+sF(t,r;y,z)+tF(r,s;z,x)=0$$ where $$F(s,t;x,y):=\sum_{k=0}^n(-1)^k\binom{s}{k}\binom{t}{n-k}B_{n-k}(x)B_k(y).$$ This symmetric relation implies the curious identities of Miki and Matiyasevich as well as some new identities for Bernoulli polynomials such as $$\sum_{k=0}^n\binom{n}{k}^2B_k(x)B_{n-k}(x)=2\sum^n\Sb k=0 k\not=n-1\endSb\binom{n}{k}\binom{n+k-1}{k}B_k(x)B_{n-k}.$$
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Finite covers of groups by cosets or subgroups
This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let $a_1G_1,...,a_kG_k$ be left cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ covers each element of $G$ at least $m$ times but none of its proper subsystems does. We show that if $G$ is cyclic, or $G$ is finite and $G_1,...,G_k$ are normal Hall subgroups of $G$, then $k\geq m+f([G:\bigcap_{i=1}^kG_i])$, where $f(\prod_{t=1}^r p_t^{α_t})=\sum_{t=1}^rα_t(p_t-1)$ if $p_1,...,p_r$ are distinct primes and $α_1,...,α_r$ are nonnegative integers. When all the $a_i$ are the identity element of $G$ and all the $G_i$ are subnormal in $G$, we prove that there is a composition series from $\bigcap_{i=1}^kG_i$ to $G$ whose factors are of prime orders. The paper also includes some other results and two challenging conjectures.
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A local-global theorem on periodic maps
Let $ψ_1,...,ψ_k$ be maps from Z to an additive abelian group with positive periods $n_1,...,n_k$ respectively. We show that the function $ψ=ψ_1+...+ψ_k$ is constant if $ψ(x)$ equals a constant for |S| consecutive integers x where S={r/n_s: r=0,...,n_s-1; s=1,...,k}; moreover, there are periodic maps $f_0,...,f_{|S|-1}$ from Z to Z only depending on S such that $ψ(x)=\sum_{r=0}^{|S|-1}f_r(x)ψ(r)$ for all integers x. This local-global theorem extends a previous result [Math. Res. Lett. 11(2004), 187--196], and has various applications.
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Restricted sumsets and a conjecture of Lev
Let A,B,S be finite subsets of an abelian group G. Suppose that the restricted sumset C={a+b: a in A, b in B, and a-b not in S} is nonempty and some c in C can be written as a+b with a in A and b in B in at most m ways. We show that if G is torsion-free or elementary abelian then |C|\geq |A|+|B|-|S| -m. We also prove that |C|\geq |A|+|B|-2|S|-m if the torsion subgroup of G is cyclic. In the case S={0} this provides an advance on a conjecture of Lev.
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A combinatorial identity with application to Catalan numbers
By a very simple argument, we prove that if $l,m,n$ are nonnegative integers then $$\sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m} =\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}. On the basis of this identity, for $d,r=0,1,2,...$ we construct explicit $F(d,r)$ and $G(d,r)$ such that for any prime $p>\max\{d,r\}$ we have \sum_{k=1}^{p-1}k^r C_{k+d}\equiv \cases F(d,r)(mod p)& if 3|p-1, \\G(d,r)\ (mod p)& if 3|p-2, where $C_n$ denotes the Catalan number $(n+1)^{-1}\binom{2n}{n}$. For example, when $p\geq 5$ is a prime, we have \sum_{k=1}^{p-1}k^2C_k\equiv\cases-2/3 (mod p)& if 3|p-1, \1/3 (mod p)& if 3|p-2; and \sum_{0<k<p-4}\frac{C_{k+4}}k \equiv\cases 503/30 (mod p)& if 3|p-1, -100/3 (mod p)& if 3|p-2. This paper also contains some new recurrence relations for Catalan numbers.
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A lower bound for |{a+b: a\in A, b\in B, P(a,b)\not=0}|
Let A and B be two finite subsets of a field F. In this paper we provide a nontrivial lower bound for |{a+b: a in A, b in B, and P(a,b) not=0}| where $P(x,y)\in F[x,y]$.
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On Snevily's conjecture and restricted sumsets
Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every positive integer m\leq (k-1)/(n-1) there are more than (k-1)n-(m+1)n(n-1)/2 sets {a_1,...,a_n} such that a_1\in A_1,..., a_n\in A_n, and both a_i\not=a_j and ma_i+b_i\not=ma_j+b_j (or both ma_i\not=ma_j and a_i+b_i\not=a_j+b_j) for all 1\leq i<j\leq n. This extends a recent result of Dasgupta, Károlyi, Serra and Szegedy on Snevily's conjecture. Actually stronger results on sumsets with polynomial restrictions are obtained in this paper.
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A sharp result on m-covers
Let A={a_s+n_sZ}_{s=1}^k be a finite system of arithmetic sequences which forms an m-cover of Z (i.e., every integer belongs at least to m members of A). In this paper we show the following sharp result: For any positive integers m_1,...,m_k and theta in [0,1), if there is a subset I of {1,...,k} such that the fractional part of sum_{s in I}m_s/n_s is theta, then there are at least 2^m such subsets of {1,...,k}. This extends an earlier result of M. Z. Zhang and an extension by Z. W. Sun. Also, we generalize the above result to m-covers of the integral ring of any algebraic number field with a power integral basis.
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On covering numbers
A positive integer n is called a covering number if there are some distinct divisors n_1,...,n_k of n greater than one and some integers a_1,...,a_k such that Z is the union of the residue classes a_1(mod n_1),...,a_k(mod n_k). A covering number is said to be primitive if none of its proper divisors is a covering number. In this paper we give some sufficient conditions for n to be a (primitive) covering number; in particular, we show that for any r=2,3,... there are infinitely many primitive covering numbers having exactly r distinct prime divisors. In 1980 P. Erdos asked whether there are infinitely many positive integers n such that among the subsets of D_n={d>1: d|n} only D_n can be the set of all the moduli in a cover of Z with distinct moduli; we answer this question affirmatively. We also conjecture that any primitive covering number must have a prime factorization p_1^{alpha_1}...p_r^{alpha_r} (with p_1,...,p_r in a suitable order) which satisfies $\prod_{0<t<s}(alpha_t+1)\ge p_s-1$ for each s=1,...,r, with strict inequality when s=r.
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