scholarly journals On a binomial coefficient and a product of prime numbers

2011 ◽  
Vol 5 (1) ◽  
pp. 87-92 ◽  
Author(s):  
Horst Alzer ◽  
Jozsef Sandor
Keyword(s):  
Prime Number ◽  
Binomial Coefficient ◽  
Prime Numbers ◽  
Double Inequality

Let Pn be the n-th prime number. We prove the following double-inequality. For all integers k?5 we have exp[k(c0?loglogk)]? k2 k/p1?p2?...?pk ? exp[k(c1?loglogk)] with the best possible constants c0 = 1/5 log23 + loglog5=1:10298? and c1 = 1/192log(36864/192)+loglog 192?1/192log(p1?p2???p192)=2.04287... This reffines a result published by Gupta and Khare in 1977.

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.

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.

On tamely ramified pro-p-extensions over -extensions of

2013 ◽  
Vol 156 (2) ◽  
pp. 281-294
Author(s):  
TSUYOSHI ITOH ◽  
YASUSHI MIZUSAWA
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Rational Number ◽  
Prime Numbers ◽  
Finite Set

AbstractFor an odd prime number p and a finite set S of prime numbers congruent to 1 modulo p, we consider the Galois group of the maximal pro-p-extension unramified outside S over the ${\mathbb Z}_p$-extension of the rational number field. In this paper, we classify all S such that the Galois group is a metacyclic pro-p group.

scholarly journals THE QUOTIENT SET OF -GENERALISED FIBONACCI NUMBERS IS DENSE IN

2017 ◽  
Vol 96 (1) ◽  
pp. 24-29 ◽  
Author(s):  
CARLO SANNA
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Algebraic Number ◽  
Fibonacci Numbers ◽  
Prime Numbers ◽  
Algebraic Number Theory ◽  
Initial Values ◽  
Quotient Set ◽  
Successive Term

The quotient set of $A\subseteq \mathbb{N}$ is defined as $R(A):=\{a/b:a,b\in A,b\neq 0\}$. Using algebraic number theory in $\mathbb{Q}(\sqrt{5})$, Garcia and Luca [‘Quotients of Fibonacci numbers’, Amer. Math. Monthly, to appear] proved that the quotient set of Fibonacci numbers is dense in the $p$-adic numbers $\mathbb{Q}_{p}$ for all prime numbers $p$. For any integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq -(k-2)}$ be the sequence of $k$-generalised Fibonacci numbers, defined by the initial values $0,0,\ldots ,0,1$ ($k$ terms) and such that each successive term is the sum of the $k$ preceding terms. We use $p$-adic analysis to generalise the result of Garcia and Luca, by proving that the quotient set of $k$-generalised Fibonacci numbers is dense in $\mathbb{Q}_{p}$ for any integer $k\geq 2$ and any prime number $p$.

scholarly journals The continuity of prime numbers can lead to even continuity

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,...

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