GOLDBACH’S CONJECTURE FROM THE PERSPECTIVE OF THE ODD PRIME FACTORS OF AN EVEN NUMBER

Abstract

In this article, Goldbach’s Strong Conjecture has been investigated. Starting from an even number N greater than 4 and not of the form “two times a prime,” a general notation has been defined to differentiate between odd prime factors of N and all the other odd primes smaller than N, which has allowed operating with them separately. Based on logical arithmetic steps with these two types of primes, two theorems have been postulated that prove that only the odd primes that are not prime factors of N can satisfy Goldbach’s Conjecture for N. Finally, a method based on the sieve of Eratosthenes has been described for generating a list of odd numbers smaller than N using only the odd prime numbers smaller than √N. Such a list has been generated for a known even number that does satisfy the Goldbach’s Conjecture. It has been concluded from this example and the theorems previously stated that such a list generated for an even number that does not satisfy the Goldbach’s Conjecture must necessarily include all odd numbers smaller than N, which would be impossible, as it would be incompatible with the natural distribution of numbers.