Related papers: Diophantine Equations Involving the Euler Totient …

We show how sums of some $5th$ powers can be written as sums of some cubics

In this article, we are interested in finding rational points on certain superelliptic curves.

Let $C_n=n2^n+1$ denote the $n$th Cullen number. There has been recent interest in finding all Cullen numbers having a given Diophantine property. We prove that, for a fixed integer $k$ and bounded integers $a_1,\ldots,a_k$, the greatest…

In this paper we propose a method of solving a Nonlinear Diophantine Equation by converting it into a System of Diophantine Linear Equations.

This is a short survey of a class of functions introduces by Tom Apostol. The survey is focused on their relation to Eulerian polynomials, derivative polynomials, and also on some integral representations.

Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the "descending power" Eulerian polynomials, and their once shifted sequence, are moment sequences for simple families of orthogonal polynomials,…

We give a closed-form expression for $\varphi(1+\varphi(2+\varphi(3+...+\varphi(n)$, where $\varphi$ is Euler's totient function. More generally, for an integer sequence $A=\{a_j\}$ we study the value of…

Some fundamental solutions of radial type for a class of iterated elliptic singular equations including the iterated Euler equation are given.

This paper investigates the exponential Diophantine equation of the form $a^x+b=c^y$, where $a, b, c$ are given positive integers with $a,c \ge 2$, and $x,y$ are positive integer unknowns. We define this form as a "Type-I transcendental…

The Euler's totient function $ \varphi(n) $ counts the positive integers up to a given integer $ n$ that are relatively prime to $ n $. We solve a problem due to Lehmer that there is no composite number $ n $ such that $ \varphi(n)\mid n-1…

We study connections between linear equations over various semigroups and recursively enumerable sets of positive integers. We give variants of the universal Diophantine representation of recursively enumerable sets of positive integers…

In this paper, by introducing the degenerate Fubini-type polynomials, we give several relations with the help of the Fa\`a di Bruno formula and some properties of Bell polynomials, and generating function methods. Also, we derive some new…

This work presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations of the form $$ \sum_{l=0}^n \binom{n}{l}…

In this paper, we deal with q-Euler numbers and q-Bernoulli numbers. We derive some interesting relations for q-Euler numbers and polynomials by using their generating function and derivative operator. Also, we show between the q-Euler…

In this paper, we find all sums of two Fibonacci numbers which are close to a power of 2. As a corollary, we also determine all Lucas numbers close to a power of 2. The main tools used in this work are lower bounds for linear forms in…

This survey paper is not a complete reference guide to number-theoretical applications of ergodic theory. Instead, it considers an approach to a class of problems involving Diophantine properties of $n$-tuples of real numbers, namely,…

Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…

We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…

Motivated by the integral representation of the Euler Beta function, we introduce its Cauchy siblings and investigate some of their properties. Two of these newly introduced functions happen to coincide with some classical means, such as…

The main idea of this article is simply calculating integer functions in module. The algebraic in the integer modules is studied in completely new style. By a careful construction, a result is proven that two finite numbers is with unequal…