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

This paper gives an heuristic lower bound for the number of integers connected to 1 and less than $x$, $\theta(x) > 0.9x,$ in the context of the $3n+1$ problem.

An algorithm is devised for computing $\Phi(n) = \phi(1) + \phi(2) + \cdots + \phi(n)$ in time $\widetilde{\Theta}(n^{2/3})$ and space $\widetilde{\Theta}(n^{1/3})$. The starting point is an existing algorithm based on the Dirichlet…

An old conjecture of Sierpinski asserts that for every integer k \ge 2, there is a number m for which the equation \phi(x)=m has exactly k solutions. Here \phi is Euler's totient function. In 1961, Schinzel deduced this conjecture from his…

Euler function $\phi(n)$ is the number of positive integers less than $n$ and relatively prime to $n$. Suppose that $\phi^1(n)=\phi(n)$ and $\phi^i(n)=\phi(\phi^{i-1}(n))$. Let $A\subseteq \mathbb{N}$, and $A_{\phi}=\{ \phi^k(n)| n\in A ,…

We propose a generic algorithm for computing the inverses of a multiplicative function under the assumption that the set of inverses is finite. More generally, our algorithm can compute certain functions of the inverses, such as their power…

We fix a positive integer $k$ and look for solutions of the equations $\phi(n+k) = \phi(n)$ and $\phi(n + k) = 2\phi(n)$. We prove that Fermat primes can be used to build five solutions for the first equation when $k$ is even and five for…

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…

We obtain a totient analogue for Linnik's theorem in arithmetic progressions. Specifically, for any coprime pair of positive integers $(m,a)$ such that $m$ is odd, there exists $n\le m^{2+o(1)}$ such that $\varphi(n)\equiv…

We present a new tool to compute the number $\phi_\A (\b)$ of integer solutions to the linear system $$ \x \geq 0 \qquad \A \x = \b $$ where the coefficients of $\A$ and $\b$ are integral. $\phi_\A (\b)$ is often described as a \emph{vector…

By using the complete discrimination system for polynomials, we study the number of positive solutions in {\small $C[0,1]$} to the integral equation {\small $\phi (x)=\int_0^1k(x,y)\phi ^n(y)dy$}, where {\small…

We show, in an effective way, that there exists a sequence of congruence classes $a_k\pmod {m_k}$ such that the minimal solution $n=n_k$ of the congruence $\phi(n)\equiv a_k\pmod {m_k}$ exists and satisfies $\log n_k/\log m_k\to\infty $ as…

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…

Lehmer's totient problem asks whether there exists any composite number $n$ such that $\varphi(n) \, \mid \, (n-1)$, where $\varphi$ is Euler totient function. It is known that if any such $n$ exists, it must be Carmichael and $n >…

In 2023, the author presented the first computable minimal Euclidean function for a non-trivial number field. Along with a formula for $\phi_{\mathbb{Z}[i]}$, the minimal Euclidean function on the Gaussian inteers, the same paper introduced…

In 1963, Edward Spence published a proof of the following With $\phi$ being Euler totient function, if $n>1$ is an integer, and if \begin{equation*} 0<a_1<\cdots<a_{\phi(n)}<n, \end{equation*} are the positive integers less than $n$,…

Every Euclidean domain $R$ has a minimal Euclidean function, $\phi_R$. A companion paper \cite{Graves} introduced a formula to compute $\phi_{\mathbb{Z}[i]}$. It is the first formula for a minimal Euclidean function for the ring of integers…

We introduce and study minimal (with respect to inclusion) solutions of systems of tropical linear differential equations. We describe the set of all minimal solutions for a single equation. It is shown that any tropical linear differential…

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 usual division algorithms on $\mathbb{Z}$ and $\mathbb{Z}[i]$ measure the size of remainders using the norm function. These rings are Euclidean with respect to several functions. The pointwise minimum of all Euclidean functions $f: R…

We obtain reasonably tight upper and lower bounds on the sum $\sum_{n \leqslant x} \varphi \left( \left\lfloor{x/n}\right\rfloor\right)$, involving the Euler functions $\varphi$ and the integer parts $\left\lfloor{x/n}\right\rfloor$ of the…