Related papers: When is 0.999... equal to 1?

Let $\{U_n\}$ be given by $U_0=1$ and $U_n=-2\sum_{k=1}^{[n/2]} \b n{2k}U_{n-2k}\ (n\ge 1)$, where $[\cdot]$ is the greatest integer function. In the paper we present a summation formula and several congruences involving $\{U_n\}$.

A natural question about Dedekind sums is to find conditions on the integers $a_1, a_2$, and $b$ such that $s(a_1,b) = s(a_2, b)$. We prove that if the former equality holds then $ b \ | \ (a_1a_2-1)(a_1-a_2)$. Surprisingly, to the best of…

We study odd numbers through a straightforward indexing. We focus in particular on odd prime and composite numbers and their distribution. With a counting argument, we calculate the limit of two sums and compare their convergence rate.

In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals

Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It is a natural…

We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.

It is a popular paradoxical exercise to show that the infinite sum of positive integer numbers is equal to -1/12, sometimes called the Ramanujan sum. Here we propose a qualitative approach, much like that of a physicist, to show how the…

The infinite numbers of the set M of finite and infinite natural numbers are defined starting from the sequence 0\Phi, where 0 is the first natural number, \Phi is a succession of symbols S and xS is the successor of the natural number x.…

We prove a sharp upper bound for the resurgence of sums of ideals involving disjoint sets of variables, strengthening work of Bisui--H\`a--Jayanthan--Thomas. Complete solutions are delivered for two conjectures proposed by these authors.…

Non-positive at infinity valuations are a class of real plane valuations which have a nice geometrical behavior. They are divided in three types. We study the dual graphs of non-positive at infinity valuations and give an algorithm for…

We prove the following results solving a problem raised in [Y. Caro, R. Yuster, On zero-sum and almost zero-sum subgraphs over $\mathbb{Z}$, Graphs Combin. 32 (2016), 49--63]. For a positive integer $m\geq 2$, $m\neq 4$, there are…

We show that there are short intervals $[x,x+y]$ containing $\gg y^{1/10}$ numbers expressible as the sum of two squares, which is many more than the average when $y=o( (\log{x})^{5/9})$. We obtain similar results for sums of two squares in…

We show that the left (right) sample quantile tends to the left (right) distribution quantile at p in [0,1], if the left and right quantiles are identical at p. We show that the sample quantiles diverge almost surely otherwise. The latter…

We consider the integers having the property of reversing when multiplied by a specific integer k. First, we proved that k should be either 1, 4 or 9. Second, we classify these integers as (10, 1)- reverse multiples, (10, 4)- reverse…

We consider the following "partition and sum" operation on a natural number: Treating the number as a long string of digits insert several plus signs in between some of the digits and carry out the indicated sum. This results in a smaller…

Let $d \geq 1$ and $s \leq 2^d$ be nonnegative integers. For a subset $A$ of vertices of the hypercube $Q_n$ and $n\geq d$, let $\lambda(n,d,s,A)$ denote the fraction of subcubes $Q_d$ of $Q_n$ that contain exactly $s$ vertices of $A$. Let…

We give two simple algorithms for the evaluation of difference between the numbers of multiples of 3 with even and odd binary digit sums in interval [0,x), and give an elementary proof of Coquet's sharp estimates for it.

We establish a new bound for the exponential sum \begin{eqnarray*} \sum_{x\in\mathcal{X}}\Big|\sum_{y\in \mathcal{Y}}\gamma(y)\exp(2\pi i a \lambda^{xy}/p)\Big|, \end{eqnarray*} where $\lambda$ is an element of the residue ring modulo a…

Given a suitable arithmetic function h, we investigate the average order of h as it ranges over the values taken by an integral binary form F. A general upper bound is obtained for this quantity, in which the dependence upon the…

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 $n\geq 1$.…