scholarly journals R-prime numbers of degree k

2018 ◽  
Vol 38 (2) ◽  
pp. 75-82
Author(s):  
Abdelhakim Chillali
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Complexity Theory ◽  
Prime Numbers ◽  
Prime Pair ◽  
Random Input ◽  
Trap Door ◽  
Twin Primes ◽  
Open Questions ◽  
One Way Function

In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the image of a random input. Here, "easy" and "hard" are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. Not being one-to-one is not considered sufficient of a function for it to be called one-way (see Theoretical Definition, in article). A twin prime is a prime number that has a prime gap of two, in other words, differs from another prime number by two, for example the twin prime pair (5,3). The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states: There are infinitely many primes p such that p + 2 is also prime. In this work we define a new notion: ‘r-prime number of degree k’ and   we give a new RSA trap-door one-way. This notion generalized a twin prime numbers because the twin prime numbers are 2-prime numbers of degree 1.

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 Some interesting tasks from the classical number theory

2020 ◽  
pp. 150-188
Author(s):  
Zurab Agdgomelashvili ◽  
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Prime Divisors ◽  
Problem Class

The article considers the following issues: – It’s of great interest for p and q primes to determine the number of those prime number divisors of a number 1 1 pq A p    that are less than p. With this purpose we have considered: Theorem 1. Let’s p and q are odd prime numbers and p  2q 1. Then from various individual divisors of the 1 1 pq A p    number, only one of them is less than p. A has at least two different simple divisors; Theorem 2. Let’s p and q are odd prime numbers and p  2q 1. Then all prime divisors of the number 1 1 pq A p    are greater than p; Theorem 3. Let’s q is an odd prime number, and p N \{1}, p]1;q] [q  2; 2q] , then each of the different prime divisors of the number 1 1 pq A p    taken separately is greater than p; Theorem 4. Let’s q is an odd prime number, and p{q1; 2q1}, then from different prime divisors of the number 1 1 pq A p    taken separately, only one of them is less than p. A has at least two different simple divisors. Task 1. Solve the equation 1 2 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 2. Solve the equation 1 3 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 3. Solve the equation 1 1 z x y p y    where p{6; 7; 11; 13;} are the prime numbers, x, y  N and y is a prime number. There is a lema with which the problem class can be easily solved: Lemma ●. Let’s a, b, nN and (a,b) 1. Let’s prove that if an  0 (mod| ab|) , or bn  0 (mod| ab|) , then | ab|1. Let’s solve the equations ( – ) in natural x , y numbers: I. 2 z x y z z x y          ; VI. (x  y)xy  x y ; II. (x  y)z  (2x)z  yz ; VII. (x  y)xy  yx ; III. (x  y)z  (3x)z  yz ; VIII. (x  y) y  (x  y)x , (x  y) ; IV. ( y  x)x y  x y , (y  x) ; IX. (x  y)x y  xxy ; V. ( y  x)x y  yx , (y  x) ; X. (x  y)xy  (x  y)x , (y  x) . Theorem . If a, bN (a,b) 1, then each of the divisors (a2  ab  b2 ) will be similar. The concept of pseudofibonacci numbers is introduced and some of their properties are found.

Preliminaries

2020 ◽  
pp. 5-56
Author(s):  
Andreas Bolfing
Keyword(s):  
Field Theory ◽  
Number Theory ◽  
Group Theory ◽  
Complexity Theory ◽  
Prime Numbers ◽  
Ring Theory ◽  
Cryptographic Primitives ◽  
Mathematical Concepts ◽  
Cyclic Groups ◽  
Efficiency Of Algorithms

Blockchains are heavily based on mathematical concepts, in particular on algebraic structures. This chapter starts with an introduction to the main aspects in number theory, such as the divisibility of integers, prime numbers and Euler’s totient function. Based on these basics, it follows a very detailed introduction to modern algebra, including group theory, ring theory and field theory. The algebraic main results are then applied to describe the structure of cyclic groups and finite fields, which are needed to construct cryptographic primitives. The chapter closes with an introduction to complexity theory, examining the efficiency of algorithms.

scholarly journals Disproof of Twin Prime Conjecture

2019 ◽  
Author(s):  
K.H.K. Geerasee Wijesuriya
Keyword(s):  
Research Paper ◽  
Prime Numbers ◽  
Prime Pair ◽  
Twin Primes ◽  
The Difference ◽  
Valid Proof

A twin prime numbers are two prime numbers which have the difference of 2 exactly. In other words, twin primes is a pair of prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research paper, my attempt is to provide a valid disproof for twin prime conjecture.

scholarly journals CARACTERIZACIÓN DE NÚMEROS PRIMOS

2017 ◽  
Vol 22 (2) ◽  
pp. 25
Author(s):  
Héctor Carlos Guimaray Huerta
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Prime Numbers ◽  
Composite Number ◽  
Palabras Clave

Los números primos es motivo de investigación en la teoría de números; en la actualidad, no existe una fórmula que nos permita obtener dichos números, y que la distribución de los mismos se considera que es aleatoria. Lo que existe son métodos para averiguar si un número es primo o compuesto. En el presente artículo se presenta una caracterización de números primos que es el complemento de los números compuestos. Palabras clave.- Divisor, Número primo, Número compuesto, Caracterización, Conjetura. ABSTRACTThe prime numbers motivate the investigation in number theory; nowadays, does not exist a formula that allows get those numbers, and that the distribution thereof is considered random. There are methods to find whether a number is prime or composite. This article presents a characterization of prime numbers which is the complement of composite numbers. Keywords.- Divisor, Prime number, Composite number, Characterization, Conjecture.

3. Prime-time mathematics

2020 ◽  
pp. 37-57
Author(s):  
Robin Wilson
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Prime Numbers ◽  
Prime Time ◽  
Leonhard Euler ◽  
Mersenne Primes ◽  
Fermat Primes

‘Prime-time mathematics’ explores prime numbers, which lie at the heart of number theory. Some primes cluster together and some are widely spread, while primes go on forever. The Sieve of Eratosthenes (3rd century BC) is an ancient method for identifying primes by iteratively marking the multiples of each prime as not prime. Every integer greater than 1 is either a prime number or can be written as a product of primes. Mersenne primes, named after French friar Marin de Mersenne, are prime numbers that are one less than a power of 2. Pierre de Fermat and Leonhard Euler were also prime number enthusiasts. The five Fermat primes are used in a problem from geometry.

scholarly journals Polynomial Characterization of Twin Primes in Function of another Prime

2013 ◽  
Vol 5 ◽  
pp. 37-39
Author(s):  
Ibrahima Gueye
Keyword(s):  
Number Theory ◽  
Prime Numbers ◽  
Twin Primes

For two millennia, the prime numbers have continued to fascinatemathematicians. Indeed, a conjecture which dates back to this period states that thenumber of twin primes is infinite. In 1949 Clement showed a theorem on twin primesIn this paper I give the proof of a polynomial characterization of twin primes usingadditive primes number theory.

scholarly journals Proof of Twin Prime Conjecture that can be obtained by using contradiction method in mathematics and it's applications to Thermal Physics

2021 ◽  
Author(s):  
K.H.K. Geerasee Wijesuriya
Keyword(s):  
Research Paper ◽  
Prime Numbers ◽  
Prime Pair ◽  
Thermal Physics ◽  
Twin Primes ◽  
The World ◽  
Subject Areas ◽  
The Difference ◽  
Three Space ◽  
Valid Proof

Twin prime numbers are two prime numbers which have the difference equals to 2 exactly. In other words, twin primes is a pair of two prime numbers which have the prime gap of exactly two. Sometimes the word “twin prime” is used for a pair of twin primes; an another name for this is considered as “prime twin” or called as “prime pair”. Up to date there is no any valid proof/disproof for twin prime conjecture since roughly more than 170 years in the world. Through this research paper, my attempt is to provide a valid proof for twin prime conjecture. This new paper is the detailed explanation of my previous paper that I completed on mid of the year 2020 titled as ‘Proof of Twin Prime Conjecture that can be obtained by using Contradiction method in Mathematics’ (WHICH IS WELL-RECONGNIZED ALL OVER THE WORLD through researchgate as well). And this proof of the existence of infinitely many twin primes can be applied to many subject areas in Physics, Chemistry and etc. And the proof of twin prime conjecture can be used to solve several unsolved problems in Physics, Chemistry and etc as well. Also as an additional result, at the end of this research paper, it discusses about an application of the Proof of Twin Prime Conjecture to the Quantum and Thermal Physics. There, this research paper consider three space volumes symbolized as area A , B and C. Inside areas A and B there are microscopic particles separately. By applying the proof of the twin prime conjecture, finally this will try to conclude that although the areas A and B have separated by area C, there are some particles those have moved from the area B to area A (due to the high thermal pressure of area B).

scholarly journals Proof of Twin Prime Conjecture that can be obtained by using Contradiction method in Mathematics

2020 ◽  
Author(s):  
K.H.K. Geerasee Wijesuriya
Keyword(s):  
Research Paper ◽  
Prime Numbers ◽  
Prime Pair ◽  
Twin Primes ◽  
The Difference ◽  
Valid Proof

Twin prime numbers are two prime numbers which have the difference of 2 exactly. In other words, twin primes is a pair of prime that has a prime gap of two. Sometimes the part “twin prime” is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research paper, my attempt is to provide a valid proof for twin prime conjecture.

scholarly journals In detail explanation for the “Proof of the Twin Prime Conjecture” and its applications to thermal physics

2021 ◽  
Author(s):  
K.H.K. Geerasee Wijesuriya
Keyword(s):  
Research Paper ◽  
Prime Numbers ◽  
Prime Pair ◽  
Thermal Physics ◽  
Twin Primes ◽  
The World ◽  
Subject Areas ◽  
The Difference ◽  
Three Space ◽  
Valid Proof

Twin prime numbers are two prime numbers which have the difference equals to exactly 2. In other words, twin primes is a pair of two prime numbers which have the value of the difference exactly two. Sometimes the word “twin prime” is used for a pair of twin primes; an another name for this is considered as “prime twin” or called as “prime pair”. Up to date there is no any exact proof/disproof for twin prime conjecture since roughly 200 years in the world. Through this research paper, my attempt is to provide a valid proof for twin prime conjecture. This new paper is the detailed explanation of my previous paper that I completed on mid of the year 2020 titled as ‘Proof of Twin Prime Conjecture that can be obtained by using Contradiction method in Mathematics’ (WHICH IS WELL-RECONGNIZED ALL OVER THE WORLD through researchgate as well). And this proof of the existence of infinitely many twin primes can be applied to many subject areas in Physics, Chemistry and etc. And the proof of twin prime conjecture can be used to solve several unsolved problems in Physics, Chemistry and etc as well. Also as an additional result, at the end of this research paper, it discusses about an application of the Proof of Twin Prime Conjecture to the Quantum and Thermal Physics. There, this research paper consider three space volumes symbolized as area A , B and C. Inside areas A and B there are microscopic particles separately. By applying the proof of the twin prime conjecture, finally this will try to conclude that although the areas A and B have separated by area C, there are some particles those have moved from the area B to area A (due to the high thermal pressure of area B).

Sign in / Sign up

Export Citation Format

Share Document