A new approach of the concept of prime number. (original) (raw)
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International Journal of Innovative Technology and Exploring Engineering, 2019
In this paper we probed some interesting aspects of primorial and factorial primes. We did some numerical analysis about the distribution of prime numbers and tabulated our findings. Also, we pointed out certain interesting facts about the utility value of the study of prime numbers and their distributions in control engineering and Brain networks.
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The paper is the ultimate prime numbers algorithm that gets rid of the unneccessary mystery about prime numbers. All the numerous arithmetic series patterns observed between various prime numbers are clearly explained with an elegant "pattern of remainders". With this algorithm we prove that odd numbers too can make an Ulam spiral contrary to current ""proofs". At the end of the paper this author proves the relationship between a simple arithmetic series pattern and the Riehmann's prime numbers distribution equation. This paper would be important for encryption too. As an example, prime integer 1979 is expressed as 1.2.4.5.10.3.7.3.1.7.26.18.11.1. This makes even smaller primes useful for encryption as well.
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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.
On the Analytical Properties of Prime Numbers
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.
Number Theory and Their Application in Computer Science and Cryptography
Journal of Mathematical Sciences & Computational Mathematics, 2020
Here we have briefly discussed the various applications of number theory in the fields of Computation with special emphasis on Encryption algorithms. We have laid special emphasis on prime numbers and briefly touched upon their importance in modern day Cryptography , especially in RSA Encryption which is the most widely used encryption technique nowadays.
Preprints, 2023
Prime number-related issues can be viewed from drastically different perspectives by examining the close connections between prime numbers and composite numbers. We think that multiple perspectives are the pillars on the path to solutions so we have created this study. As a result of the study, we proposed two new formulas by presenting three theorems and one proof for each theorem, a total of three proofs. We proved that the formula p • n + p returns a composite number in the first of the theorems, which is the preliminary theorem. Our first theorem except the preliminary theorem is that the formula p • n + p returns all composite numbers, and we proved that too. Finally, we created Theorem II using Theorem I to use in our other work and proved that the formula 2 • n • p + p returns all odd composite numbers, which is Theorem II. Afterward, we presented the similarities of the 2 • n • p + p formula we put forth with another known formula.
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Engineering and Applied Science Letters, 2020
Prime numbers and their variations are extremely useful in applied research areas such as cryptography, feedback and control in engineering. In this paper we discuss about prime numbers, perfect numbers, even perfect and odd perfect numbers, amicable numbers, semiprimes, mersenne prime numbers, triangular numbers, distribution of primes, relation between π and prime numbers. In the process we also obtain interesting properties of some of them and raise a set of open problems for further exploration.
The implications of Prime numbers and the Riemann hypothesis on Asymmetric Cryptography
2020
The academic study of prime numbers has been of mathematical interest for centuries and over time remarkable progress has been made in understanding the unique properties and patterns of these numbers. Over the last fifty years, the discovery of mathematical models has aided the progression of computer science. Whilst encryption, previously used for communication in the wars, has now been adopted into quotidian life. Mathematicians have discovered new methods for the secure transmission of information and have augmented them by introducing new messaging platforms using encryption algorithms based on prime numbers. In this dissertation, the importance of prime numbers and their application to asymmetric cryptographic systems will be outlined. Furthermore, it will be shown how effective modern-day public-key cryptographic systems are, based on a coded model of the RSA algorithm. Moreover, it will be evident why the Riemann Hypothesis could encode the best possible prediction of the di...
On the location and classification of all prime numbers
Arxiv preprint arXiv:0707.1041, 2007
Abstract: We will describe an algorithm to arrange all the positive and negative integer numbers. This array of numbers permits grouping them in six different Classes, alpha\ alpha alpha, beta\ beta beta, gamma\ gamma gamma, delta\ delta delta, epsilon\ epsilon epsilon, and zeta\ zeta zeta. Particularly, numbers belong to ...