a power digraph, denoted by $g(n,k)$, is a directed graph with $z_{n}={0,1,..., n-1}$ as the set of vertices and $l={(x,y):x^{k}equiv y~(bmod , n)}$ as the edge set, where $n$ and $k$ are any positive integers. in this paper, the structure of $g(2q+1,k)$, where $q$ is a sophie germain prime is investigated. the primality tests for the integers of the form $n=2q+1$ are established in terms of th...

The upper bound and the lower bound of average numbers of divisors of Euler Phi function and Carmichael Lambda function are obtained by Luca and Pomerance (see [LP]). We improve the lower bound and provide a heuristic argument which suggests that the upper bound given by [LP] is indeed close to the truth.

For g, n coprime integers, let `g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of `g(n) as n ≤ x ranges over integers coprime to g, and x tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptot...

For g, n coprime integers, let `g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of `g(n) as n ≤ x ranges over integers coprime to g, and x tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptot...

We assign to each positive integer n a digraph Γ(n, 11) whose set of vertices is Zn = {0, 1, 2, ..., − 1} and there exists exactly one directed edge from b if only a11 ≡ b(mod n), where a, ∈ Zn. Let Γ1(n, be the subdigraph induced by which are coprime n. discuss when regular or semi-regular. A formula for number fixed points established. necessary sufficient condition symmetry proved. Moreover,...

We describe an algorithm for constructing Carmichael numbers N with a large number of prime factors p1, p2, . . . , pk. This algorithm starts with a given number Λ = lcm(p1 − 1, p2 − 1, . . . , pk − 1), representing the value of the Carmichael function λ(N). We found Carmichael numbers with up to 1101518 factors.

Giuga has conjectured that if the sum of the (n− 1)-st powers of the residues modulo n is −1 (mod n), then n is 1 or prime. It is known that any counterexample is a Carmichael number. Lehmer has asked if φ(n) divides n−1, with φ being Euler’s function, must it be true that n is 1 or prime. No examples are known, but a composite number with this property must be a Carmichael number. We show that...

A characteriation of continuity of the $p$-$Lambda$-variation function is given and the Helly's selection principle for $Lambda BV^{(p)}$ functions is established. A characterization of the inclusion of Waterman-Shiba classes into classes of functions with given integral modulus of continuity is given. A useful estimate on modulus of variation of functions of class $Lambda BV^{(p)}$ is found.