Related papers: On Differences of Multiplicative Functions and Sol…

We study the distribution of solutions of equations $\sigma(n)=\sigma(n+k)$ and $\varphi(n)=\varphi(n+k)$. We give new upper bounds for these solutions.

Erd\H{o}s and Graham define $g(n) = n + \phi(n)$ and the iterated application $g_k(n) = g(g_{k-1}(n))$. They ask for solutions of $g_{k+r}(n) = 2 g_{k}(n)$ and observe $g_{k+2}(10) = 2 g_{k}(10)$ and $g_{k+2}(94) = 2 g_{k}(94)$. We show…

Let $n_0$ be 1 or 3. If a multiplicative function $f$ satisfies $f(p+q-n_0) = f(p)+f(q)-f(n_0)$ for all primes $p$ and $q$, then $f$ is the identity function $f(n)=n$ or a constant function $f(n)=1$.

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…

In this paper, we are going to describe the solutions of the functional equation $$ \varphi\Big(\frac{x+y}{2}\Big)(f(x)+f(y))=\varphi(x)f(x)+\varphi(y)f(y) $$ concerning the unknown functions $\varphi$ and $f$ defined on an open interval.…

We examine how closely a multiplicative function resembles an additive function. Given a multiplicative function $g$ and an additive function $f$, we examine the size of the quantity $E(f,g;x)=\# \{n\leq x:f(n)=g(n)\}$. We establish a lower…

In this article we prove that equation $\phi(x)=n$, for a fixed $n$, admits a finite number of solutions, we find the general form of these solutions, and we show that: if $x_0$ is a unique solution of this equation then $x_0$ is a product…

In this sequence of work we investigate polynomial equations of additive functions. We consider the solutions of equation \[ \sum_{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x)^{q_{i}}= 0 \qquad \left(x\in \mathbb{F}\right), \] where $n$ is a positive…

In this note we investigate the solutions of certain meta-Fibonacci recurrences of the form $f(n)=f(n-f(n-1))+f(n-2)$ for various sets of initial conditions. In the case when $f(n)=1$ for $n\leq 1$, we prove that the resulting integer…

It is well-known that the congruence $\sum_{i=1}^{ n} i^{ n} \equiv 1 \pmod{n}$ has exactly five solutions: $\{1,2,6,42,1806\}$. In this work, we characterize the solutions to the congruence $1^n + 2^n + \dotsb + n^n\equiv p \pmod{n}$ for…

In this paper, we consider a variant of Pillai's problem over function fields $ F $ in one variable over $ \mathbb{C} $. For given simple linear recurrence sequences $ G_n $ and $ H_m $, defined over $ F $ and satisfying some weak…

We deduce asymptotic formulas for the alternating sums $\sum_{n\le x} (-1)^{n-1} f(n)$ and $\sum_{n\le x} (-1)^{n-1} \frac1{f(n)}$, where $f$ is one of the following classical multiplicative arithmetic functions: Euler's totient function,…

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…

In this paper we consider $N$, the number of solutions $(x,y,u,v)$ to the equation $ (-1)^u r a^x + (-1)^v s b^y = c$ in nonnegative integers $x, y$ and integers $u, v \in \{0,1\}$, for given integers $a>1$, $b>1$, $c>0$, $r>0$ and $s>0$.…

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…

We consider $N$, the number of solutions $(x,y,u,v)$ to the equation $ (-1)^u r a^x + (-1)^v s b^y = c $ in nonnegative integers $x, y$ and integers $u, v \in \{0,1\}$, for given integers $a>1$, $b>1$, $c>0$, $r>0$ and $s>0$. When…

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…

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…

We consider the equation $[p_{1}^{c}] + [p_{2}^{c}] + [p_{3}^{c}] = N$, where $N$ is a sufficiently large integer, and prove that if $1 < c < \frac{17}{16}$, then it has a solution in prime numbers $p_{1}$, $p_{2}$, $p_{3}$ such that each…

Power functions with low $c$-differential uniformity have been widely studied not only because of their strong resistance to multiplicative differential attacks, but also low implementation cost in hardware. Furthermore, the…