Factors of Fermât Numbers and Large Primes of the Form k • 2 " + 1
نویسندگان
چکیده
In addition, a factor of F75 discovered by Gary Gostin is presented. The current status for all Fm is shown in a table. Primes of the form k ■ 2" + I, k odd. are listed for 31 =£ k < 149, 1500 < n « 4000, and for 151 « k « 199, 1000 < n « 4000. Some primes for even larger values of n are included, the largest one being 5 • 213'65 + 1. Also, a survey of several related questions is given. In particular, values of A such that A • 2" + 1 is composite for every n are considered, as well as odd values of h such that 3/i • 2" ± 1 never yields a twin prime pair.
منابع مشابه
The power digraphs of safe primes
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...
متن کاملApplications of Multizeta Values to Mahler Measure
These notes correspond to a mini-course taught by the author during the program “PIMS-SFU undergraduate summer school on multiple zeta values: combinatorics, number theory and quantum field theory”. Please send any comments or corrections to the author at [email protected]. 1. Primes, Mahler Measure, and Lehmer’s question We start our study by discussing prime numbers. First consider the ...
متن کاملPrimality Tests for Numbers of the Form
Let k,m ∈ Z, m ≥ 2, 0 < k < 2 and 2 6| k. In the paper we give a general primality criterion for numbers of the form k·2±1, which can be viewed as a generalization of the LucasLehmer test for Mersenne primes. In particular, for k = 3, 9 we obtain explicit primality tests, which are simpler than current known results. We also give a new primality test for Fermat numbers and criteria for 9 · 2 ± ...
متن کاملAsymptotic behaviour of associated primes of monomial ideals with combinatorial applications
Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$. We say that $I$ satisfies the persistence property if $mathrm{Ass}_R(R/I^k)subseteq mathrm{Ass}_R(R/I^{k+1})$ for all positive integers $kgeq 1$, which $mathrm{Ass}_R(R/I)$ denotes the set of associated prime ideals of $I$. In this paper, we introduce a class of square-free monomial ideals in the polynomial ring $R=K[x_1,ld...
متن کاملOn the Fermat Periods of Natural Numbers
Let b > 1 be a natural number and n ∈ N0. Then the numbers Fb,n := b 2 +1 form the sequence of generalized Fermat numbers in base b. It is well-known that for any natural number N , the congruential sequence (Fb,n (mod N)) is ultimately periodic. We give criteria to determine the length of this Fermat period and show that for any natural number L and any b > 1 the number of primes having a peri...
متن کامل