Related papers: On several irrationality problems for Ahmes series

We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed $m$ there are at most $\mathcal{O}_{\epsilon}(n^{3/5+\epsilon})$ solutions of…

Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. The concept of the weighted sum $\sum_{n\in{\rm NR}}\lambda^{n}$ is introduced in \cite{KZ0,KZ}, where ${\rm NR}={\rm NR}(a_1,a_2,\dots,a_k)$ denotes the set of…

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…

Given $n$ positive integers $a_1,a_2,\dots,a_n$, and a positive integer right hand side $\beta$, we consider the feasibility version of the subset sum problem which is the problem of determining whether a subset of $a_1,a_2,\dots,a_n$ adds…

In this paper, we consider rational hypergeometric series of the form \[\frac{p}{\pi}= \sum_{k=0}^\infty u_k\quad\text{with}\quad u_k=\frac{\left(\frac{1}{2}\right)_k \left(q\right)_k \left(1-q\right)_k}{(k!)^3}(r+s\,k)\,t^k,\] where…

A_k = {1, a_2, ..., a_k} is an h-basis for n if every positive integer not exceeding n can be expressed as the sum of no more than h values a_i; we write n = n_h(A_k). An extremal h-basis A_k is one for which n is as large as possible, and…

For any set $A$ of natural numbers with positive upper Banach density, we show the existence of an infinite set $B$ and sequences $(t_k)_{k\in \mathbb{N}}, (s_k)_{k\in \mathbb{N}}$ of natural numbers such that $\left\{ \sum_{n \in F}n : F…

A subset of an abelian group is {\em sequenceable} if there is an ordering $(x_1, \ldots, x_k)$ of its elements such that the partial sums $(y_0, y_1, \ldots, y_k)$, given by $y_0 = 0$ and $y_i = \sum_{j=1}^i x_i$ for $1 \leq i \leq k$, are…

Noam Elkies and Everett Howe independently noticed a certain elegant product formula for the multiple integral \int_R \prod_{1 \le i < j \le k} (x_j-x_i) dx_1 \cdots dx_k, where the region $R$ is the set of $k$-tuples satisfying $0 < x_1 <…

For any real-valued $k > 1$, we consider the tree rooted at 0, where each positive integer $n$ has parent $\lfloor\frac{n}{k}\rfloor$. The average number of children per node is $k$, thus this definition gives a natural way to extend…

A classical argument was introduced by Khintchine in 1926 in order to exhibit the existence of totally irrational singular linear forms in two variables. This argument was subsequently revisited and extended by many authors. For instance,…

We answer a number of questions of Erd\H{o}s on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a…

For $n=1,2,3,\ldots$ let $S_n$ be the sum of the first $n$ primes. We mainly show that the sequence $a_n=\root n\of{S_n/n}\ (n=1,2,3,\ldots)$ is strictly decreasing, and moreover the sequence $a_{n+1}/a_n\ (n=10,11,\ldots)$ is strictly…

We consider a class of double exponential sums studied in a paper of Sinai and Ulcigrai. They proved a linear bound for these sums along the sequence of denominators in the continued fraction expansion of $\alpha$, provided $\alpha$ is…

Let $k$ be a field. Let $A=\prod_{i=1}^r K_i$ and $B=\prod_{j=1}^s E_j$ be \'etale $k$-algebras where $K_i$ and $E_j$ are finite separable field extensions of $k$ with $[K_i:k]=m_i$ and $[E_j:k]=n_j$. Let…

A sequence of non-negative integers is called a B_k sequence if all the sums of arbitrary k elements are different. In this paper, we will present a new estimation for the upper bound of B_k sequences.

We consider series of the form $$ \frac{p}{q} +\sum_{j=2}^\infty \frac{1}{x_j}, $$ where $x_1=q$ and the integer sequence $(x_n)$ satisfies a certain non-autonomous recurrence of second order, which entails that $x_n|x_{n+1}$ for $n\geq 1$.…

We obtained the region of convergence and the summation formula for some modified generalized hypergeometric series (1.2). We also investigated rationality of the sums of the power series (1.3). As a result the series (1.4) cannot be the…

Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question…

We explicitly describe a noteworthy transcendental continued fraction in the field of power series over Q, having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set {1, 2}.…