Carmichael Numbers of the form ( 6 m + 1 ) ( 12 m + 1 ) ( 18 m + 1 ) Harvey Dubner 449
نویسنده
چکیده
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.
منابع مشابه
Carmichael numbers in the sequence (2n k+1)n≥1
A Carmichael number is a positive integer N which is composite and the congruence aN ≡ a (mod N) holds for all integers a. The smallest Carmichael number is N = 561 and was found by Carmichael in 1910 in [6]. It is well– known that there are infinitely many Carmichael numbers (see [1]). Here, we let k be any odd positive integer and study the presence of Carmichael numbers in the sequence of ge...
متن کاملRepunit R49081 is a probable prime
The Repunit R49081 = (1049081 − 1)/9 is a probable prime. In order to prove primality R49080 must be approximately 33.3% factored. The status of this factorization is included. Repunits are numbers of the form Rn = (10 − 1)/9. Considerable computer time has been spent factoring repunits, creating interesting primes based on the characteristics of repunits, and searching for repunit primes. One ...
متن کاملDistribution of generalized Fermat prime numbers
Numbers of the form Fb,n = b 2n+1 are called Generalized Fermat Numbers (GFN). A computational method for testing the probable primality of a GFN is described which is as fast as testing a number of the form 2m−1. The theoretical distributions of GFN primes, for fixed n, are derived and compared to the actual distributions. The predictions are surprisingly accurate and can be used to support Ba...
متن کامل1 99 8 Higher - Order Carmichael Numbers Everett
We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZ-algebra that can be generated as a Z/nZ-module by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdős for the usual Carmichael numbers) that indic...
متن کاملFinding strong pseudoprimes to several bases II
Define ψm to be the smallest strong pseudoprime to all the first m prime bases. If we know the exact value of ψm, we will have, for integers n < ψm, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψm are known for 1 ≤ m ≤ 8. Upper bounds for ψ9, ψ10 and ψ11 were first given by Jaeschke, and those for ψ10 and ψ11 were...
متن کامل