Related papers: Generalizing the Lehmer's totient problem

Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ \varphi(n)$ be the Euler totient function. The asymptotic formula for the new finite sum over the primes $ \sum_{p\leq…

Let $n$ be a positive integer and $G(n)$ denote the number of non-isomorphic finite groups of order $n$. It is well-known that $G(n) = 1$ if and only if $(n,\phi(n)) = 1$, where $\phi(n)$ and $(a, b)$ denote the Euler's totient function and…

Let $\varphi(n)$ denote the Euler totient function. In this paper, we first establish a new upper bound for $n/\varphi(n)$ involving $K(n)$, the function that counts the number of primorials not exceeding $n$. In particular, this leads to…

Euler totient function $\phi(n)$ plays a central role in number theory and is applied in areas such as cryptography. In this paper, we study iterations of the totient function. We first prove that for any integer $n>2$, iteratively applying…

In 2000 Deaconescu raised a question whether there exists a composite $n$ for which $S_2(n)|\phi(n)-1$, where $\phi(n)$ is Euler's function and $S_2(n)$ is Schemmel's totient function. In this paper we prove that any such $n$ is odd,…

We show that for some $k\le 3570$ and all $k$ with $442720643463713815200|k$, the equation $\phi(n)=\phi(n+k)$ has infinitely many solutions $n$, where $\phi$ is Euler's totient function. We also show that for a positive proportion of all…

Inspired by Lehmer's and Deaconescu's conjectures, as well as various analogue problems concerning Euler's totient function $\varphi(n)$, Schemmel's totient function $S_{2}(n)$, Jordan totient function $J_k$, and the unitary totient…

In this note we generalize and prove a recent conjecture of Varchenko concerning the number of critical points of a (multivalued) meromorphic function $\phi$ on an algebraic manifold. Under certain conditions, this number turns out to…

Considering $\mathbb{Z}_n$ the ring of integers modulo $n$, the classical Fermat-Euler theorem establishes the existence of a specific natural number $\varphi(n)$ satisfying the following property: $ x^{\varphi(n)}=1%\hspace{1.0cm}\text{for…

For a function $f\colon \mathbb{N}\to\mathbb{N}$, define $N^{\times}_{f}(x)=\#\{n\leq x: n=kf(k) \mbox{ for some $k$} \}$. Let $\tau(n)=\sum_{d|n}1$ be the divisor function, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and…

This thesis is about the study of Diophantine equations involving binary recurrent sequences with arithmetic functions. Various Diophantine problems are investigated and new results are found out of this study. Firstly, we study several…

Let $b>1$ be an odd positive integer and $k, l \in \mathbb{N}$. In this paper, we show that every positive rational number can be written as $\varphi(m^{2})/(\varphi(n^{2}))^{b}$ and $\varphi(k(m^{2}-1))/\varphi(ln^{2})$, where $m, n\in…

Let $\phi(n)$ be the Euler totient function and $\sigma(n)$ denote the sum of divisors of $n$. In this note, we obtain explicit upper bounds on the number of positive integers $n\leq x$ such that $\phi(\sigma(n)) > cn$ for any $c>0$. This…

Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. The aim of this article is to give a result about the sum of euler's totient function from k equal 1 to n whene p divides n and p…

We study the set D of positive integers d for which the equation $\phi(a)-\phi(b)=d$ has infinitely many solution pairs (a,b), where $\phi$ is Euler's totient function. We show that the minumum of D is at most 154, exhibit a specific A so…

The aim of this note is to provide an upper bound of the number of positive integers $\le x$ which can be written as $\varphi(n)$ for some positive integer $n$, where $\varphi$ stands for the Euler's function. The order of magnitude of this…

Motivated by an application of semigroup variants to the discrete log problem in groups and related cryptographic applications, we introduce a new kind of totient function, related to both Euler's function and a generalisation of Euler's…

For a function $f\colon \mathbb{N}\to\mathbb{N}$, let $$ N^+_f(x)=\{n\leq x: n=k+f(k) \mbox{ for some } k\}. $$ Let $\tau(n)=\sum_{d|n}1$ be the divisor function, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and…

The purpose of this article is to introduce the concept of invariance and its properties. These properties can be used to check the primality of a number. Combining these properties with the Euler theorem, it is possible to generalize this…

We prove that neither a prime nor {an l-almost prime} number theorem hold in the class of regular Toeplitz subshifts. But, {when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler's totient…