Related papers: A Solution of Sierpinski Problem Based m

Refining an estimate of Croot, Dobbs, Friedlander, Hetzel and Pappalardi, we show that for all $k \geq 2$, the number of integers $1 \leq a \leq n$ such that the equation $a/n = 1/m_1 + \dotsc + 1/m_k$ has a solution in positive integers…

This paper investigates coefficients of cyclotomic polynomials theoretically and experimentally. We prove the following result. {{\em If $n=p_1\ldots p_k$ where $p_i$ are odd primes and $p_1<p_2<\ldots<p_r<p_1+p_2<p_{r+1}<\ldots<p_t$ with…

A k-composition of n is a sequence of length k of positive integers summing up to n. In this paper, we investigate the number of k-compositions of n satisfying two natural coprimality conditions. Namely, we first give an exact asymptotic…

A classical result of A. Fleck states that if p is a prime, and n>0 and r are integers, then $$\sum_{k=r(mod p)}\binom {n}{k}(-1)^k=0 (mod p^{[(n-1)/(p-1)]}).$$ Recently R. M. Wilson used Fleck's congruence and Weisman's extension to…

Let $k$ be an integer which is the difference between prime numbers infinitely often. It is known that there are infinitely many such $k$ and, in this paper, we give a new unconditional proof that these $k$ have positive density and improve…

Ellipsephic or Kempner-like harmonic series are series of inverses of integers whose expansion in base $B$, for some $B \geq 2$, contains no occurrence of some fixed digit or some fixed block of digits. A prototypical example was proposed…

Let $m>2$ and $q>0$ be integers with $m$ even or $q$ odd. We show the supercongruence $$\sum_{k=0}^{p-1}(-1)^{km}\binom{p/m-q}{k}^m\equiv0\pmod{p^3}.$$ for any prime $p>mq$. This confirms a conjecture of Sun.

Agust\'{i}n-Aquino solved, in terms of the table of marks of $\Aff(\mathbb{Z}/2k\mathbb{Z})$, the problem of enumerating the classes of bicolour self-complementary and rigid patterns in $\mathbb{Z}/2k\mathbb{Z}$ (also known as \emph{strong…

We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd's and lcm's of $k$ positive integers, where $k\ge 2$ is fixed. We refine and generalize an asymptotic formula of Bordell\`{e}s (2007),…

In 2022, Z.-W. Sun defined \begin{equation*} w_k^{(\alpha)}{(x)}=\sum_{j=1}^{k}w(k,j)^{\alpha}x^{j-1}, \end{equation*} where $k,\alpha$ are positive integers and $w(k,j)=\frac{1}{j}\binom{k-1}{j-1}\binom{k+j}{j-1}$. Let $(x)_{0}=1$ and…

We present a novel conjecture concerning the additive representation of natural numbers using prime powers. Based on extensive computational verification, we conjecture that every integer n > 23 can be expressed as a sum of at most five…

It is conjectured that for a perfect number $m,$ $\rm{rad}(m)\ll m^{\frac{1}{2}}.$ We prove bounds on the radical of multiperfect number $m$ depending on its abundancy index. Assuming the ABC conjecture, we apply this result to study gaps…

A classical theorem by Jacobson says that a ring in which every element $x$ satisfies the equation $x^n=x$ for some $n>1$ is commutative. According to Birkhoff's Completeness Theorem, if $n$ is fixed, there must be an equational proof of…

In this paper we study recurrences concerning the combinatorial sum $[n,r]_m=\sum_{k\equiv r (mod m)}\binom {n}{k}$ and the alternate sum $\sum_{k\equiv r (mod m)}(-1)^{(k-r)/m}\binom{n}{k}$, where m>0, $n\ge 0$ and r are integers. For…

By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integers $S_k(n) = 1^k + 2^k + \cdots + n^k$, we derive a couple of infinite families of explicit formulas for $S_k(n)$. One of the families…

In this paper we construct a cover {a_s(mod n_s)}_{s=1}^k of Z with odd moduli such that there are distinct primes p_1,...,p_k dividing 2^{n_1}-1,...,2^{n_k}-1 respectively. Using this cover we show that for any positive integer m divisible…

Let $n$ be a non-negative integer and put $p_{n}(x)=\prod_{i=0}^{n}(x+i)$. In the first part of the paper, for given $n$, we study the existence of integer solutions of the Diophantine equation $$ y^m=p_{n}(x)+\sum_{i=1}^{k}p_{a_{i}}(x), $$…

Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $\alpha$ ($\leq k$), let \[\Sigma_{\alpha}…

Let $I_k = [(2k-1)^2, (2k+1)^2)$ for $k \geq 1$. Starting from the odd-composite matrix $(b_{ij})$ with $b_{ij} = (2i-1)(2j-1)$, introduced by the author in [1], we define for each odd integer $n$ the \emph{matrix multiplicity} $r(n)$, the…

Let $r\ge k\ge 2$ be fixed positive integers. Let $\varrho_{r,k}$ denote the characteristic function of the set of $r$-tuples of positive integers with $k$-wise relatively prime components, that is any $k$ of them are relatively prime. We…