We have constructed a Carmichael number with 10,333,229,505 prime factors, and have also constructed Carmichael numbers with k prime factors for every k between 3 and 19,565,220. These computations are the product of implementations of two new algorithms for the subset product problem that exploit the non-uniform distribution of primes p with the property that p − 1 divides a highly composite Λ.

Numbers of the form (6m+1)(12m+1)(18m+1) where all three factors are simultaneously prime are the best known examples of Carmichael numbers. In this paper we tabulate the counts of such numbers up to 10 for each n ≤ 42. We also derive a function for estimating these counts that is remarkably accurate.

We investigate two arithmetic functions naturally occurring in the study of the Euler and Carmichael quotients. The functions are related to the frequency of vanishing of the Euler and Carmichael quotients. We obtain several results concerning the relations between these functions as well as their typical and extreme values.

Let φ and λ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that φ(λ(n)) = λ(φ(n)). We also study the normal order of the function φ(λ(n))/λ(φ(n)).

Let φ and λ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that φ(λ(n)) = λ(φ(n)). We also study the normal order of the function φ(λ(n))/λ(φ(n)).

This study presents primality testing by mostly emphasizing on probabilistic primality tests. It is proposed an algorithm for the generation and testing of primes, and explained Carmichael numbers. The importance of prime numbers in encryption is stated and experimental results are given. KeywordsPrime Numbers, Carmichael Numbers, Primality Tests, Prime Generation, Security Protocols, Factoriza...

We show, in an effective way, that there exists a sequence of congruence classes ak (mod mk) such that the minimal solution n = nk of the congruence φ(n) ≡ ak (mod mk) exists and satisfies log nk/ logmk → ∞ as k → ∞. Here, φ(n) is the Euler function. This answers a question raised in [3]. We also show that every congruence class containing an even integer contains infinitely many values of the ...