Related papers: The Smallest Solution of \phi(30n+1)<\phi(30n) is …

We consider iterations of integer-valued functions $\phi$, which have no fixed points in the domain of positive integers. We define a local function $\phi_n$, which is a sub-function of $\phi$ being restricted to the subdomain $\{0, ..., n…

Diophantine equations can often be reduced to various types of classical Thue equations. These equations usually have only very small solutions, on the other hand to compute all solutions (i.e. to prove the non-existence of large solutions)…

The prime-counting function $\pi(x)$ which computes the number of primes smaller or equal to a given real number has a long-standing interest in number theory. The present manuscript proposes a method to compute $\pi(x)$ with time…

Let $f(x)$ be a polynomial of degree $n \ge 1$ with real coefficients and let $X \ge 2$ and $\delta \ge 0$ be real numbers. Let $\|\cdot\|$ be the distance to the nearest integer. We obtain upper bounds for the number of solutions to the…

The primorial $p\#$ of a prime $p$ is the product of all primes $q\le p$. Let pr$(n)$ denote the largest prime $p$ with $p\# \mid \phi(n)$, where $\phi$ is Euler's totient function. We show that the normal order of pr$(n)$ is $\log\log…

The present work includes some of the author's original researches on integer solutions of Diophantine liner equations and systems. The notion of "general integer solution" of a Diophantine linear equation with two unknowns is extended to…

The sufficient conditions for insolvability of the Diophantine equation $\sum_{i=1}^{m}x_i^{n}=bc^{n}$ ($n, m \geq 2$, $b, c\in \mathbb{N}$) in nonnegative integers are obtained for the case where the canonical decomposition of the number…

We study existence, structure, uniqueness and regularity of solutions of the obstacle problem \begin{equation*} \inf_{u\in BV_f(\Omega)}\int_{\mathbb{R}^n}\phi(x,Du), \end{equation*} where $BV_f(\Omega)=\{u\in BV(\Omega): u\geq \psi \text{…

In this work, we develop a new iterative method for computing the digits of $\pi$ by argument reduction of the tangent function. This method combines a modified version of the iterative formula for $\pi$ with squared convergence that we…

Let $c$ be fixed with $1<c<35/34$. In this paper we prove that for every sufficiently large real number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*}…

A partition of a positive integer $n$ is a non-increasing sequence of positive integers which sum to $n$. A recently studied aspect of partitions is the minimal excludant of a partition, which is defined to be the smallest positive integer…

A vanishing sum of roots of unity is called minimal if no proper, nonempty sub-sum of it vanishes. This paper classifies all minimal vanishing sums of roots of unity of weight at most 16 by hand, thereby uncovering new phenomena beyond the…

The $3x+1$ Problem asks if whether for every natural number $n$, there exists a finite number of iterations of the piecewise function $$f(2n)=n, \quad f(2n-1)=6n-2, $$ with an iterate equal to the number $1$, or in other words, every…

A property $\Pi$ on a finite set $U$ is \emph{monotone} if for every $X \subseteq U$ satisfying $\Pi$, every superset $Y \subseteq U$ of $X$ also satisfies $\Pi$. Many combinatorial properties can be seen as monotone properties. The problem…

A $3$-partition of an $n$-element set $V$ is a triple of pairwise disjoint nonempty subsets $X,Y,Z$ such that $V=X\cup Y\cup Z$. We determine the minimum size $\varphi_3(n)$ of a set $\mathcal{E}$ of triples such that for every 3-partition…

We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and…

A classical result of Honsberger states that the number of incongruent triangles with integer sides and perimeter $n$ is the nearest integer to $\frac{n^2}{48}$ ($n$ even) or $\frac{(n+3)^2}{48}$ ($n$ odd). We solve the analogous problem…

Lehmer's totient problem consists of determining the set of positive integers $n$ such that $\varphi(n)|n-1$ where $\varphi$ is Euler's totient function. In this paper we introduce the concept of $k$-Lehmer number. A $k$-Lehmer number is a…

We develop a new closed-form arithmetic and recursive formula for the partition function and a generalization of Andrews' smallest parts (spt) function. Using the inclusion-exclusion principle, we additionally develop a formula for the…

Let $G=(V,E)$ be a simple, connected graph. One is often interested in a short path between two vertices $u,v$. We propose a spectral algorithm: construct the function $\phi:V \rightarrow \mathbb{R}_{\geq 0}$ $$ \phi = \arg\min_{f:V…