Related papers: On common divisors of multinomial coefficients

In 2004, Zhi-Wei Sun posed the following conjecture: If a_1G_1,...,a_kG_k (k>1) are finitely many pairwise disjoint left cosets in a group G with all the indices [G:G_i] finite, then for some 1\le i<j\le k, the greatest common divisor of…

For any integer $r \geq 1$, the sequence of numbers $\{{c^{(r)}_{k}}\}_{k \geq 0} $ is defined implicitly by [\sum_k\binom{n}{k}^r\binom{n+k}{k}^r = \sum_k\binom{n}{k}\binom{n+k}{k}c^{(r)}_k,\quad n=0,1,2,...] Asmus Schmidt conjectured that…

The Alder-Andrews Theorem, a partition inequality generalizing Euler's partition identity, the first Rogers-Ramanujan identity, and a theorem of Schur to $d$-distinct partitions of $n$, was proved successively by Andrews in 1971, Yee in…

The author advocates two specific mathematical notations from his popular course and joint textbook, "Concrete Mathematics". The first of these, extending an idea of Iverson, is the notation "[P]" for the function which is 1 when the…

Two numbers $m$ and $n$ are considered amicable if the sum of their proper divisors, $s(n)$ and $s(m)$, satisfy $s(n) = m$ and $s(m) = n$. In 1981, Pomerance showed that the sum of the reciprocals of all such numbers, $P$, is a constant. We…

Arrondo, Sols and De Cataldo proved that there are only finitely many families of codimension two subvarieties not of general type in the smooth quadric of dimension $n+2$ for $n\ge 2 $, $n\neq 4$. In this paper we drop the assumption…

Menger's Edge Theorem asserts that there exist $k$ pairwise edge-disjoint paths between two vertices in an undirected graph if and only if a deletion of any $k-1$ or less edges does not disconnect these two vertices. Alternatively, there…

For all positive integers n, we prove the following divisibility properties: $$(2n+3){2n\choose n} | 3{6n\choose 3n}{3n\choose n}, and (10n+3){3n\choose n} | 21{15n\choose 5n} {5n\choose n}.$$ This confirms two recent conjectures of Z.-W.…

An interesting, and still wide open, conjecture of Reiner and Stanton predicts that certain "strange" symmetric differences of $q$-binomial coefficients are always nonnegative and unimodal. We extend their conjecture to a broader, and…

Let $S_k(m):=1^k+2^k+\cdots+(m-1)^k$ denote a power sum. In 2011 Bernd Kellner formulated the conjecture that for $m\ge 4$ the ratio $S_k(m+1)/S_k(m)$ of two consecutive power sums is never an integer. We will develop some techniques that…

Subresultants of two univariate polynomials are one of the most classic and ubiquitous objects in computational algebra and algebraic geometry. In 1948, Habicht discovered and proved interesting relationships among subresultants. Those…

Let $\mathcal{A}_1,\ldots,\mathcal{A}_m$ be families of $k$-subsets of an $n$-set. Suppose that one cannot choose pairwise disjoint edges from $s+1$ distinct families. Subject to this condition we investigate the maximum of…

In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X^2+nY^2. Based on some numerical computations, they observed that the…

We consider polynomials of the form t^n-1 and determine when members of this family have a divisor of every degree in Z[t]. With F(x) defined to be the number of such integers up to x, we prove the existence of two positive constants c_1…

For $n=0,1,2,\ldots$ let $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k}$. In this paper we illustrate the connection between $\{d_n^{(r)}(x)\}$ and Meixner polynomials. New formulas and recurrence relations for $d_n^{(r)}(x)$ are…

Let $F(x)$ be an irreducible polynomial with integer coefficients and degree at least 2. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y<d\le z$. We…

In 1992, Erd$\H{o}$s and Hegyv$\'{a}$ri showed that for any prime p, there exist infinitely many length 3 weakly prime-additive numbers divisible by p. In 2018, Fang and Chen showed that for any positive integer m, there exists infinitely…

Let $\{a_1, . . . , a_n\}$ be a set of positive integers with $a_1 < \dots < a_n$ such that all $2^n$ subset sums are distinct. A famous conjecture by Erd\H{o}s states that $a_n>c\cdot 2^n$ for some constant $c$, while the best result known…

In order to give a unified generalization of the BW inequality and the DDVV inequality, Lu and Wenzel proposed three Conjectures 1, 2, 3 and an open Question 1 in 2016. In this paper we discuss further these conjectures and put forward…

We show that the greatest prime factor of $n^2+h$ is at least $n^{1.312}$ infinitely often. This gives an unconditional proof for the range previously known under the Selberg eigenvalue conjecture. Furthermore, we get uniformity in $h \leq…