Related papers: Mean Divisibility of Multinomial coefficients

We prove a version of Gauss's Lemma. It recursively constructs polynomials {c_k} for k=0,1,...,m+n, in Z[a_i,A_i,b_j,B_j] for i=0,...,m, and j=0,1,...,n, having degree at most (m+n choose m) in each of the four variable sets, such that…

Let $a(n, k)$ be the $k$-th coefficient of the $n$-th cyclotomic polynomial. Recently, Ji, Li and Moree \cite{JLM09} proved that for any integer $m\ge1$, $\{a(mn, k)| n, k\in\mathbb{N}\}=\mathbb{Z}$. In this paper, we improve this result…

The integers $n=\prod_{i=1}^r p_i^{a_i}$ and $m=\prod_{i=1}^r p_i^{b_i}$ having the same prime factors are called exponentially coprime if $(a_i,b_i)=1$ for every $1\le i\le r$. We estimate the number of pairs of exponentially coprime…

Given a natural number $n \geq 4$ we show that there exists infinitely many polynomials $f_{n}(x):= \prod_{i=1}^{n} (x^{2} - a_{i})$ such that (i) $f_{n}(x)$ has a root modulo every positive integer, (ii) $f_{n}(x)$ has no rational roots,…

Given an integer $k$, define $C_k$ as the set of integers $n > \max(k,0)$ such that $a^{n-k+1} \equiv a \pmod{n}$ holds for all integers $a$. We establish various multiplicative properties of the elements in $C_k$ and give a sufficient…

Let $m$, $r$ and $n$ be positive integers. We denote by ${\bf k}\vdash n$ any tuple of odd positive integers ${\bf k}=(k_1,\dots,k_t)$ such that $k_1+\dots+k_t=n$ and $k_j\ge 3$ for all $j$. In this paper we prove that for every…

We consider a joint ordered multifactorisation for a given positive integer $n\geq 2$ into $m$ parts, where $n=n_1~\times~\ldots~\times~n_m$, and each part $n_j$ is split into one or more component factors. Our central result gives an…

For a sequence $M=(m_{i})_{i=0}^{\infty}$ of integers such that $m_{0}=1$, $m_{i}\geq 2$ for $i\geq 1$, let $p_{M}(n)$ denote the number of partitions of $n$ into parts of the form $m_{0}m_{1}\cdots m_{r}$. In this paper we show that for…

In the present paper we initiate the study of a certain kind of partition inequality, by showing, for example, that if $M\geq 5$ is an integer and the integers $a$ and $b$ are relatively prime to $M$ and satisfy $1\leq a<b<M/2$, and the…

Let $\{U_n\}_{n\geq 0}$ be a Lucas sequence. Then the equation $$|U_n|=m_1!m_2!\cdots m_k!$$ with $1<m_1\leq m_2\leq \cdots\leq m_k$ implies $n\in \{1,2, 3, 4, 6, 8, 12\}$. Further the equation $$|U_n|=D_{m_1}D_{m_2}\cdots D_{m_k}, \qquad…

For Lucas sequences of the first kind (u_n) and second kind (v_n) defined as usual for positive n by u_n=(a^n-b^n)/(a-b), v_n=a^n+b^n, where a and b are either integers or conjugate quadratic integers, we describe the set of indices n for…

A sequence $s(n)$ of integers is MC-finite if for every $m \in \mathbb{N}^+$ the sequence $s^m(n) = s(n) \bmod{m}$ is ultimately periodic. We discuss various ways of proving and disproving MC-finiteness. Our examples are mostly taken from…

Let $n$ be a positive integer and $f(x) := x^{2^n}+1$. In this paper, we study orders of primes dividing products of the form $P_{m,n}:=f(1)f(2)\cdots f(m)$. We prove that if $m > \max\{10^{12},4^{n+1}\}$, then there exists a prime divisor…

Let $m\ge 2$ be a fixed positive integer. Suppose that $m^j \leq n< m^{j+1}$ is a positive integer for some $j\ge 0$. Denote $b_{m}(n)$ the number of $m$-ary partitions of $n$, where each part of the partition is a power of $m$. In this…

Let p(n, k) denote the number of partitions of n into parts less than or equal to k. We show several properties of this function modulo 2. First, we prove that for fixed positive integers k and m, p(n,k) is periodic modulo m. Using this, we…

Let n be a non-null positive integer and $d(n)$ is the number of positive divisors of n, called the divisor function. Of course, $d(n) \leq n$. $d(n) = 1$ if and only if $n = 1$. For $n > 2$ we have $d(n) \geq 2$ and in this paper we try to…

Fix positive reals $a,b,c,d$, and let $h(x)$ be a real function behaving sort of like $\sin x$ near 0. Then, provided $m$ grows linearly with $n$. there exists a positive constant $C$ such that$$…

We denote by $c_t^{(m)}(n)$ the coefficient of $q^n$ in the series expansion of $(q;q)_\infty^m(q^t;q^t)_\infty^{-m}$, which is the $m$-th power of the infinite Borwein product. Let $t$ and $m$ be positive integers with $m(t-1)\leq 24$. We…

Let $p \equiv 1 \pmod{4}$ be prime, let $m$ and $n$ be integers such that $p=m^2+n^2$, and let $b$ be a positive integer. Let $Q(z,q) = (z,q/z,q;q)_{\infty}(qz^2,q/z^2;q^2)_{\infty}$ denote the product appearing in the quintuple product…

Let $N$ be any fixed positive integer and define \begin{align*} S_N(x)=\sum_{m, n \leq x}d(n^2+Nm^2), \end{align*} where $d(n)$ is the divisor function. We evaluate asymptotically $S_N(x)$ for several $N$, extending earlier works of Gafurov…