On Prime number varieties and their applications (original) (raw)
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On Prime Numbers and Related Applications
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.
On Problems Related to Primes: Some Ideas
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.
2021
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.
On the statistical distribution of prime numbers
2017
Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation among the natural numbers is an ancient dilemma. The properties of the functions 2ab+a+b in the domain of natural numbers are introduced, analyzed, and exhibited to illustrate how these single out all the prime numbers from the full set of odd numbers. The characterization of odd primes vs. odd non-primes can be done with 2ab+a+b among the odd natural numbers as an analogue to the other, well known type of fundamental characterization for irrational and rational numbers among the real numbers. The prime number theorem, twin primes and erratic nature of primes, are also commented upon with respect to selection, as well as with the Fermat and Euler numbers as examples.
Two hundred and thirteen conjectures on primes
In two of my previous published books, “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, respectively “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function”, I already expressed my passion for integer numbers, especially for primes and Fermat pseudoprimes, fascinating numbers that seem to be a little bit more willing to let themselves ordered and understood than the prime numbers. This book brings together sixty-two papers on prime numbers, many of them supporting the author’s belief, expressed above, namely that new ordered patterns can be discovered in the “undisciplined” set of prime numbers, observing the ordered patterns in the set of Fermat pseudoprimes, especially in the set of Carmichael numbers, the absolute Fermat pseudoprimes, and in the set of Poulet (sometimes also called Sarrus) numbers, the relative Fermat pseudoprimes to base two. Other papers, which are not based on the observation of Fermat pseudoprimes, are based on the observation of Mersenne numbers, Fermat numbers, Smarandache generalized Fermat numbers, and other well known or less known classes of integers which are very much related with the study of primes. Part One of this book of collected papers contains two hundred and thirteen conjectures on primes and Part Two of this book brings together the articles regarding primes, submitted by the author to the preprint scientific database Vixra, representing the context of the conjectures listed in Part One, papers regarding squares of primes, semiprimes, twin primes, sequences of primes, types of duplets or triplets of primes, special classes of composites, ways to write primes, formulas for generating large primes, generalizations of the twin primes and de Polignac’s conjecture, generalizations of Cunningham chains and Fermat numbers and many other classic issues regarding prime numbers. Finally, in the last eight from these collected papers, I defined a new function, the MC function, and I showed some of its possible applications (for instance, I conjectured that for any pair of twin primes p and p + 2, where p ≥ 5, there exist a positive integer n of the form 15 + 18*k such that the value of Smarandache function for n is equal to p and the value of MC function for n is equal to p + 2, I also made a Diophantine analysis of few Smarandache type sequences using the MC function).
2017
Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation among the natural numbers is an ancient dilemma. The properties of the functions 2ab+a+b in the domain of natural numbers are introduced, analyzed, and exhibited to illustrate how these single out all the prime numbers from the full set of odd numbers. The characterization of odd primes vs. odd non-primes can be done with 2ab+a+b among the odd natural numbers as an analogue to the other, well known type of fundamental characterization for irrational and rational numbers among the real numbers. The prime number theorem, twin primes and erratic nature of primes, are also commented upon with respect to selection, as well as with the Fermat and Euler numbers as examples. Keywords prime number generator, prime number theorem, twin primes, erratic nature of primes
Some New Notes on Mersenne Primes and Perfect Numbers
Indonesian Journal of Mathematics Education, 2020
Mersenne primes are a specific type of prime number that can be derived using the formula M_p=2^p - 1, where p is a prime number. A perfect number is a positive integer of the form P(p)=2^(p-1)(2^p - 1) where 2^p - 1 is a Mersenne prime and can be written as the sum of its proper divisor, that is, a number which is half the sum of all of its positive divisor. In this paper, some concepts relating to Mersenne primes and perfect numbers were revisited. Mersenne primes and perfect numbers were evaluated using triangular numbers. Further, this paper discussed how to partition perfect numbers into odd cubes for odd prime The formula that partition perfect numbers in terms of its proper divisors were developed. The results of this study are useful to understand the mathematical structures of Mersenne primes and perfect numbers.
Special Primes And Some Of Their Properties
2020
In this paper, we present the denition, some properties and solve a problem on special primes. These properties help in providing us with better understanding of the problem posed related to special primes on the open problem garden website [1].The problem involves nding all the primes q, given a prime p such that q ≡ 1(modp) and 2 q1 p ≡ 1(modq). We prove that a prime number q is a special prime of p if and only if order of 2 in U(q) divides q1 p . Also we prove that a prime number q is not a special prime for any prime number if 2 is a generator of the group U(q) and some other properties.