A Natural Partial Order on The Prime Numbers

Acta Mathematica Hungarica, 1993

Let (N; ) be the multiplicative semigroup of the natural numbers endowed with the degree mapping de…ned by the recurrence (p (k)) = 1 + (k), where p (k) denotes the k-th prime number for all k 1. If N # (k) and P # N (k) denote, respectively, the cardinal of the set of all naturals and all primes with degree k, it is shown that, as k ! 1, P # (k)=N # (k) converges to the Otter constant 0:338 32185689:::: Thus, asymptotically, among all naturals with the same degree, about one third are prime.

1, 2022

Since the dawn of time, man has been organizing space and time by means of numbers, but I will try to present how the prime numbers themselves are ordered in this book. I will present prime numbers in a new light, in the light of their basic properties. What are these basic properties of prime numbers, we will see on the example of numbers from 1 to 9, where there are as many as four (2,3,5,7), which are, how they arise and in what order they follow each other. I further explain how the prime numbers are woven into the sequence of natural numbers, revealing all their hitherto hidden beauty, which is a reflection of the order, beauty and truth encoded in them for centuries.

We present some new ideas on important problems related to primes. The topics of our discussion are: simple formulae for primes, twin primes, Sophie Germain primes, prime tuples less than or equal to a predefined number, and their infinitude; establishment of a kind of similarity between natural numbers and numbers that appear in an arithmetic progression, similar formulae for primes and the so called generalized twin primes in an arithmetic progression and their infinitude; generalization of Bertrand postulate and a Bertrand like postulate for twin primes; some elementary implications of a simple primality test, the use of Chinese remainder theorem in a possible proof of the Goldbach conjecture; Schinzel Sierpinski conjecture; and lastly the Mersenne primes and composites, Fermat primes, and their infinitude.

IntechOpen's , 2023

In this work we have studied the prime numbers in the model P ¼ am þ 1, m, a>1∈ . and the number in the form q ¼ mam þ bm þ 1 in particular, we provided tests for hem. This is considered a generalization of the work José María Grau and Antonio M. Oller-marcén prove that if Cmð Þ¼ a mam þ 1 is a generalized Cullen number then ma m - ð Þ1 a ð Þ mod Cmð Þ a . In a second paper published in 2014, they also presented a test for Broth’s numbers in Form kpn þ 1 where k<p n . These results are basically a generalization of the work of W. Bosma and H.C Williams who studied the cases, especially when p ¼ 2, 3, as well as a generalization of the primitive MillerRabin test. In this study in particular, we presented a test for numbers in the form mam þ bm þ 1 in the form of a polynomial that highlights the properties of these numbers as well as a test for the Fermat and Mersinner numbers and p ¼ ab þ 1 a, b>1∈  and p ¼ qa þ 1 where q is prime odd are special cases of the number mam þ bm þ 1 when b takes a specific value. For example, we proved if p ¼ qa þ 1 where q is odd prime and a>1∈  where πj ¼ 1 q q j
 then Pq2 j¼1 πjð Þ Cmð Þ a qj1 q  a m ð Þ - χð Þ m,qam ð Þ mod p Components of proof Binomial the- orem Fermat’s Litter Theorem Elementary algebra.

2007

For integers a and b we define the Shanks chain p1 ; p2 ; : : : ; pk of length k to be a sequence of k primes such that p i+1 = ap i 2 \Gamma b for i = 1; 2; : : : ; k \Gamma 1. While for Cunningham chains it is conjectured that arbitrarily long chains exist, this is, in general, not true for Shanks chains. In fact, with s = ab we show that for all but 56 values of s 1000 any corresponding Shanks chain must have bounded length. For this, we study certain properties of functional digraphs of quadratic functions over prime fields, both in theory and practice. We give efficient algorithms to investigate these properties and present a selection of our experimental results.

Advances in Pure Mathematics, 2014

The main design of this paper is to determine once and for all the true nature and status of the sequence of the prime numbers, or primes-that is,

2016

The fundamental theorem of arithmetic states that any composite natural integer can be expressed in one and only one way as a product of prime numbers. This sets the understanding of the organization of prime numbers at the core of number theory. In this work we present a simple, self-consistent and deterministic scheme allowing to investigate further the intrinsic organization of prime numbers. Using this scheme, we establish an algorithm that yields the complete list of prime numbers below any preassigned limit x. Counting the latter yields π(x), the number of prime numbers below x. Based on preliminary tests on computing clusters available, a considerable gain in computational speed and algorithmic simplicity towards producing complete lists of large prime numbers is observed. At the core of the new scheme lays its ability to provide, in a deterministic way, complete lists of consecutive and composite odd numbers below any preassigned limit x. The complete list of prime numbers b...

In this paper, we generalize the concept of prime number and define the real primes. It allows to apply the new concept to cryptology.

Lecture Notes in Mathematics, 1985

In this paper we show how Galois theory for rings can be applied to the problem of distinguishing prime numbers from composite numbers. It develops ideas that were first formulated in [11, Section 8; 12]. A positive integer n is prime if and only if the ring ZZ/nZZ, is a field. Many primality testing algorithms make use of extension rings A of 2Z/nE that are fields if n is prime. They depend on known properties of such fields and of the Frobenius map A-> A that sends every χ e A to its n-th power. If n is composite then usually one of these properties is found not to be satisfied, and one is finished. If one does not succeed in proving n composite in this way then the problem suggests itself how to prove that n is prime. Only after this proof has been completed one knows that the rings one works with are actually fields; in particular, this fact may not be used in the proof. It is for this reason that Galois theory for rings rather than for fields is needed. Galois theory for rings can be found in [4; 6, Chapter III]. For the convenience of the reader we prove in Section 2 all facts from this theory that we need, starting only from basic properties of tensor products, localizations, and projective modules [1; 2]. In Section 3 we restrict to finite rings and abelian Galois groups, and we treat the Artin symbol, which replaces the Frobenius map. Section 4 is devoted to a special class of extensions of Z3/nE, which we call cyclotomic extensions. These play an important role in primality testing. In Section 5 we prove a result about Gauss sums that can be viewed äs a generalization of [5, Theorem (7.8)], and we show how to Interpret this result in terms of Artin symbols. The application to primality testing occupies Section 6. We describe a test that is closely related to the methods of [3], äs generalized by Williams (see [14] for references). The second test that we describe is an improvement of the method proposed in [5]. Finally, we show how the theory presented in this paper can be used to combine the two tests. It may be expected that this combined method, once implemented, will perfonn better than any existing primality testing algorithm.