In my first paper on the expansion to Dirichilet's theorem, the "Exemption Rule" allowed for us t... more In my first paper on the expansion to Dirichilet's theorem, the "Exemption Rule" allowed for us to identify which terms in an arithmetic progression described by an+b, where gcd(a,b)=1, would be prime. In the RSA algorithm, we require semi-prime numbers, which are products of two prime numbers p and q, where p and q can be either distinct or identical. The exemption rule allows us to generate n-values that would lead an+b to be of the form P(2k+1), a product of two odd numbers, and the exemption rule ensures that the two numbers are prime. To further enhance the security of the algorithm, at the end of the process, we can implement dynamic mapping functions which adds another layer of complexity, and with machine learning capabilities, create a modern enigma machine, in which the configurations are changing periodically.
An Expansion To Dirchlet's Theorem: Exemption Rule, 2023
Prime numbers are interesting numbers as they are very essential to our daily life, like they are... more Prime numbers are interesting numbers as they are very essential to our daily life, like they are used in encryption of our messages so that we can send secure texts to our loved ones. However, there is not any arithmetic structure to them, thus, making applications involving prime numbers very difficult and tedious. Luckily, our predecessors created few sieving techniques to filter out prime numbers: Sieve of Eratosthene is one of the most known one, where you list down all the positive integers from 1 to n, and cancel out all the even numbers, followed by all the multiples of 3, so on and so forth for the rest of multiples of prime numbers, and eventually you are left with just prime numbers. Though it is an easy technique to acquire prime numbers, for large n, it becomes tedious. Generally, a prime sieve works by removing composite numbers, which is generated by the sieve technique applied, to leave behind just prime numbers. However, sieving methods do not work to find individual prime numbers,and give rise to structure for prime numbers. Therefore, this paper will explore a new sieving technique called the 'Exemption Rule', , which is an expansion to Dirchlet's theorem on infinitely many primes in an = 2 + − arithmetic progression (AP), an+b, where gcd(a,b) is 1. This rule filters out composite numbers from any of these AP by exempting n values that result in an+b being composite, and leave behind just prime numbers, while also giving arithmetic structure for the prime number,p=an+b, for some n value.
The paper explores the relationship between primes in arithmetic progressions and Goldbach's Conj... more The paper explores the relationship between primes in arithmetic progressions and Goldbach's Conjecture, which posits that every even integer greater than 4 can be expressed as the sum of two prime number
The central idea for Goldbach's Conjecture in the paper is to express even integers greater than 4 as 2(N+3) for N≥0, using 2n+3 as an expression for primes.
The paper delves into the concept of "Exemption Rule" in detail, explaining how it sieves out composite terms in the arithmetic progression, 2n+3, and identifies prime terms based on the odd multiples of the prime P.
In my first paper on the expansion to Dirichilet's theorem, the "Exemption Rule" allowed for us t... more In my first paper on the expansion to Dirichilet's theorem, the "Exemption Rule" allowed for us to identify which terms in an arithmetic progression described by an+b, where gcd(a,b)=1, would be prime. In the RSA algorithm, we require semi-prime numbers, which are products of two prime numbers p and q, where p and q can be either distinct or identical. The exemption rule allows us to generate n-values that would lead an+b to be of the form P(2k+1), a product of two odd numbers, and the exemption rule ensures that the two numbers are prime. To further enhance the security of the algorithm, at the end of the process, we can implement dynamic mapping functions which adds another layer of complexity, and with machine learning capabilities, create a modern enigma machine, in which the configurations are changing periodically.
An Expansion To Dirchlet's Theorem: Exemption Rule, 2023
Prime numbers are interesting numbers as they are very essential to our daily life, like they are... more Prime numbers are interesting numbers as they are very essential to our daily life, like they are used in encryption of our messages so that we can send secure texts to our loved ones. However, there is not any arithmetic structure to them, thus, making applications involving prime numbers very difficult and tedious. Luckily, our predecessors created few sieving techniques to filter out prime numbers: Sieve of Eratosthene is one of the most known one, where you list down all the positive integers from 1 to n, and cancel out all the even numbers, followed by all the multiples of 3, so on and so forth for the rest of multiples of prime numbers, and eventually you are left with just prime numbers. Though it is an easy technique to acquire prime numbers, for large n, it becomes tedious. Generally, a prime sieve works by removing composite numbers, which is generated by the sieve technique applied, to leave behind just prime numbers. However, sieving methods do not work to find individual prime numbers,and give rise to structure for prime numbers. Therefore, this paper will explore a new sieving technique called the 'Exemption Rule', , which is an expansion to Dirchlet's theorem on infinitely many primes in an = 2 + − arithmetic progression (AP), an+b, where gcd(a,b) is 1. This rule filters out composite numbers from any of these AP by exempting n values that result in an+b being composite, and leave behind just prime numbers, while also giving arithmetic structure for the prime number,p=an+b, for some n value.
The paper explores the relationship between primes in arithmetic progressions and Goldbach's Conj... more The paper explores the relationship between primes in arithmetic progressions and Goldbach's Conjecture, which posits that every even integer greater than 4 can be expressed as the sum of two prime number
The central idea for Goldbach's Conjecture in the paper is to express even integers greater than 4 as 2(N+3) for N≥0, using 2n+3 as an expression for primes.
The paper delves into the concept of "Exemption Rule" in detail, explaining how it sieves out composite terms in the arithmetic progression, 2n+3, and identifies prime terms based on the odd multiples of the prime P.
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Paper by Chockalingam Kumarappan
The central idea for Goldbach's Conjecture in the paper is to express even integers greater than 4 as 2(N+3) for N≥0, using 2n+3 as an expression for primes.
The paper delves into the concept of "Exemption Rule" in detail, explaining how it sieves out composite terms in the arithmetic progression, 2n+3, and identifies prime terms based on the odd multiples of the prime P.
The central idea for Goldbach's Conjecture in the paper is to express even integers greater than 4 as 2(N+3) for N≥0, using 2n+3 as an expression for primes.
The paper delves into the concept of "Exemption Rule" in detail, explaining how it sieves out composite terms in the arithmetic progression, 2n+3, and identifies prime terms based on the odd multiples of the prime P.