scholarly journals Prime Numbers Distribution Line

2020 ◽  
Author(s):  
Shcherbakov Aleksandr Gennadiyevich
Keyword(s):  
Prime Number ◽  
Prime Factor ◽  
Prime Numbers ◽  
The Third ◽  
Distribution Line ◽  
Natural Series

During the analysis of the fractal-primorial periodicity of the natural series of numbers, presented in the form of an alternation (sequence) of prime numbers (1 smallest prime factor > 1 of any integer), the regularity of prime numbers distribution was revealed. That is, the theorem is proved that for any integer = N on the segment of the natural series of numbers from 1 to N + 2N: (1) prime numbers are arranged in groups, by exactly three consecutive prime numbers of the form: (Р1-Р2-Р3). In this case, the distance from the first to the third prime number of any group is less than 2N integers, that is, Р3–Р1 < 2N integers. (2) These same prime numbers are redistributed in a line in groups, by exactly two consecutive prime numbers, on all segments of the natural series of numbers shorter than 2Nintegers.

scholarly journals On the Concept of Representations of Prime Numbers and Prime Products

2018 ◽  
Vol 10 (1) ◽  
pp. 132
Author(s):  
Peter Bissonnet
Keyword(s):  
Prime Number ◽  
Double Helix ◽  
Prime Numbers ◽  
The Third ◽  
Triangular Representation

The author used to think that the only representation of prime numbers, etc. was the prime number double helix representation. However, in 2011, the author published a paper which puzzled him and it eventually dawned upon him that there was more than one representation of prime numbers. The second representation will be referred to as the hyperbolic representation. The third representation will be called the parabolic representation and is related to the hyperbolic representation. The fourth representation is the triangular representation and is related to the hyperbolic representation. Both of the primary representations, the double helix and the hyperbolic both reject that 2 and 3 are prime numbers. The hyperbolic representation is shown to be related to Lorentz-like transformations.

scholarly journals The continuity of prime numbers can lead to even continuity(The ultimate)

2021 ◽  
Author(s):  
Xie Ling
Keyword(s):  
Prime Number ◽  
Prime Numbers ◽  
Original Group

Abstract n continuous prime numbers can combine a group of continuous even numbers. If an adjacent prime number is followed, the even number will continue. For example, if we take prime number 3, we can get even number 6. If we follow an adjacent prime number 5, we can get even numbers by using 3 and 5: 6, 8 and 10. If a group of continuous prime numbers 3,5,7,11,... P, we can get a group of continuous even numbers 6,8,10,12,..., 2n. Then if an adjacent prime number q is followed, the original group of even numbers 6,8,10,12,..., 2n will be finitely extended to 2 (n + 1) or more adjacent even numbers. My purpose is to prove that the continuity of prime numbers will lead to even continuity as long as 2 (n + 1) can be extended. If the continuity of even numbers is discontinuous, it violates the Bertrand Chebyshev theorem of prime numbers. Because there are infinitely many prime numbers: 3, 5, 7, 11,... We can get infinitely many continuous even numbers: 6,8,10,12,...

scholarly journals The continuity of prime numbers can lead to even continuity(The ultimate)

2021 ◽  
Author(s):  
Xie Ling
Keyword(s):  
Prime Number ◽  
Prime Numbers ◽  
Original Group

Abstract n continuous prime numbers can combine a group of continuous even numbers. If an adjacent prime number is followed, the even number will continue. For example, if we take prime number 3, we can get even number 6. If we follow an adjacent prime number 5, we can get even numbers by using 3 and 5: 6, 8 and 10. If a group of continuous prime numbers 3,5,7,11,... P, we can get a group of continuous even numbers 6,8,10,12,..., 2n. Then if an adjacent prime number q is followed, the original group of even numbers 6,8,10,12,..., 2n will be finitely extended to 2 (n + 1) or more adjacent even numbers. My purpose is to prove that the continuity of prime numbers will lead to even continuity as long as 2 (n + 1) can be extended. If the continuity of even numbers is discontinuous, it violates the Bertrand Chebyshev theorem of prime numbers. Because there are infinitely many prime numbers: 3, 5, 7, 11,... We can get infinitely many continuous even numbers: 6,8,10,12,...

scholarly journals RSA algorithm using key generator ESRKGS to encrypt chat messages with TCP/IP protocol

2020 ◽  
Vol 8 (2) ◽  
pp. 113-120
Author(s):  
Aminudin Aminudin ◽  
Gadhing Putra Aditya ◽  
Sofyan Arifianto
Keyword(s):  
Prime Number ◽  
Instant Messaging ◽  
Prime Numbers ◽  
Public Key ◽  
Key Generation ◽  
Security Testing ◽  
Rsa Algorithm ◽  
Computer Resource ◽  
Encryption And Decryption ◽  
Private And Public

This study aims to analyze the performance and security of the RSA algorithm in combination with the key generation method of enhanced and secured RSA key generation scheme (ESRKGS). ESRKGS is an improvement of the RSA improvisation by adding four prime numbers in the property embedded in key generation. This method was applied to instant messaging using TCP sockets. The ESRKGS+RSA algorithm was designed using standard RSA development by modified the private and public key pairs. Thus, the modification was expected to make it more challenging to factorize a large number n into prime numbers. The ESRKGS+RSA method required 10.437 ms faster than the improvised RSA that uses the same four prime numbers in conducting key generation processes at 1024-bit prime number. It also applies to the encryption and decryption process. In the security testing using Fermat Factorization on a 32-bit key, no prime number factor was found. The test was processed for 15 hours until the test computer resource runs out.

scholarly journals Asymptotic counting theorems for primitive juggling patterns

2019 ◽  
Vol 15 (05) ◽  
pp. 1037-1050
Author(s):  
Erik R. Tou
Keyword(s):  
Prime Number ◽  
Generating Functions ◽  
Arbitrary Number ◽  
Mathematical Theory ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Theoretical Framework ◽  
Positive Integers ◽  
Infinite Set ◽  
Fixed Period

The mathematics of juggling emerged after the development of siteswap notation in the 1980s. Consequently, much work was done to establish a mathematical theory that describes and enumerates the patterns that a juggler can (or would want to) execute. More recently, mathematicians have provided a broader picture of juggling sequences as an infinite set possessing properties similar to the set of positive integers. This theoretical framework moves beyond the physical possibilities of juggling and instead seeks more general mathematical results, such as an enumeration of juggling patterns with a fixed period and arbitrary number of balls. One problem unresolved until now is the enumeration of primitive juggling sequences, those fundamental juggling patterns that are analogous to the set of prime numbers. By applying analytic techniques to previously-known generating functions, we give asymptotic counting theorems for primitive juggling sequences, much as the prime number theorem gives asymptotic counts for the prime positive integers.

scholarly journals On a sequence of prime numbers

1968 ◽  
Vol 8 (3) ◽  
pp. 571-574 ◽  
Author(s):  
C. D. Cox ◽  
A. J. Van Der Poorten
Keyword(s):  
Prime Factor ◽  
Prime Numbers ◽  
Image Position

Euclid's scheme for proving the infinitude of the primes generates, amongst others, the following sequence defined by p1 = 2 and pn+1 is the highest prime factor of p1p2…pn+1.

About finding of prime numbers that follows after given prime number without using computer

2020 ◽  
pp. 81-85
Author(s):  
V. S. Malakhovsky
Keyword(s):  
Prime Number ◽  
Prime Numbers ◽  
Arithmetic Progressions

It is shown how to define one or several prime numbers following af­ter given prime number without using computer only by calculating sev­eral arithmetic progressions. Five examples of finding such prime num­bers are given.

Investigating Prime Numbers and the Great Internet Mersenne Prime Search

2002 ◽  
Vol 8 (2) ◽  
pp. 70-76
Author(s):  
Jeffrey J. Wanko ◽  
Christine Hartley Venable
Keyword(s):  
Middle School ◽  
Middle School Students ◽  
Prime Number ◽  
Prime Numbers ◽  
School Students ◽  
Large Numbers ◽  
Mersenne Prime

Middle school students learn about patterns, formulas, and large numbers motivated by a search for the largest prime number. Activities included.

scholarly journals XXVIII.—On a Proposition in the Theory of Numbers

1857 ◽  
Vol 21 (3) ◽  
pp. 407-409
Author(s):  
Balfour Stewart
Keyword(s):  
Prime Number ◽  
Prime Factor ◽  
Prime Factors ◽  
Theory Of Numbers

Problem. If p be one of the roots of the equation xm − 1 = 0, (not 1,) then (1 − p)(1 − p2) …. (1 − pm − 1) = m, provided m is a prime number.If m be not a prime number, and if , the same will hold for all roots p = p1α, where α is a number < m and prime to m. But for all roots p = p1α, where α, or one of its prime factors, is also a prime factor of m, the product (1 − p)(1 − p2) …. (1 − pm − 1) will be equal to 0.

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