Some characterizations of twin prime numbers and their application

Abstract

Letting Bertrand’s Postulate, Goldbach’s Conjecture and Chen’s theorem as base theorems, we find some new results. This research work proposes a theorem that every even number greater than 6 can be written as a sum of two primes, where at least one is a member of a twin prime. In addition, we propose a lemma which states that for any positive integer n, 2n>12has at least one twin prime in between n and 2n. In this study, we establish two propositions. First proposition finds out few prime summations of two consecutive even integers if the prime summation of one is known and where one prime is one of a twin prime. Second proposition finds out the equality of two primes from the middle of the integer. The construction of the distance 6m+4 between two consecutive twin primes, where m is any nonnegative integer, is shown and illustrated by examples. The distance between two consecutive twin primes is calculated by congruence modulo. All those results are verified by programming language Python. Graphical representation of the distance between two consecutive twin primes is presented in this research by MS Excel. There are a lot of different types of applications of primes in mathematics, science, engineering and technology. Among them, prime numbers used in cryptography has changed revolutionarily the whole secret communication system and greatly developed the cyber security system. This thesis work briefly enlighten on cryptography and two old models of cryptography, namely, Diffie-Hellman public key cryptography and RSA cryptosystem as applications of primes.