Let φ denote the Euler function. In this paper, we estimate the number of positive integers n ≤ x with the property that if a prime p > y divides φ(n), then p2 ∤ φ(n). We also give similar estimates for the Carmichael function λ.

In this paper, we prove Hyers-Ulam-Rassias stability of $C^*$-ternary algebra homomorphism for the following generalized Cauchy-Jensen equation $$eta mu fleft(frac{x+y}{eta}+zright) = f(mu x) + f(mu y) +eta f(mu z)$$ for all $mu in mathbb{S}:= { lambda in mathbb{C} : |lambda | =1}$ and for any fixed positive integer $eta geq 2$ on $C^*$-ternary algebras by using fixed poind alternat...

We establish several related results on Carmichael, Sierpiński and Riesel numbers. First, we prove that almost all odd natural numbers k have the property that 2nk + 1 is not a Carmichael number for any n ∈ N; this implies the existence of a set K of positive lower density such that for any k ∈ K the number 2nk + 1 is neither prime nor Carmichael for every n ∈ N. Next, using a recent result of ...

Fermat wrote in a letter to Frenicle, that whenever p is prime, p divides a p?1 ? 1 for all integers a not divisible by p, a result now known as Fermat's `little theorem'. An equivalent formulation is the assertion that p divides a p ? a for all integers a, whenever p is prime. The question naturally arose as to whether the primes are the only integers exceeding 1 that satisfy this criterion, b...

An important unsolved question in number theory is the Lehmer's totient problem that asks whether there exists any composite $n$ such $\varphi(n)\mid n-1$, where $\varphi$ Euler's function. It known if exists, it must be odd, square-free, greater $10^{30}$, and divisible by at least $15$ distinct primes. Such a also Carmichael number. In this short note, we discuss group-theoretical analogous...

This paper proves that if a Carmichael number is composed of primes pi, then the LCM of the pi − 1’s can never be of the form 2 and can be of the form 2 ∗P for only four prime values of P ≤ 127. Moreover, if the LCM of a Carmichael number is 2 ∗ P and a Carmichael number m has all of its Fermat prime factors among the five known Fermat primes, then P must be one of the four possibilities mentio...