Secret of prime numbers and the Goldbach Conjecture

Abstract

This paper presents an empirical study of a theorem about prime numbers that is little studied: "Every prime number greater than 3 equals a multiple of 6 increased or decreased by one unit." This long-proven theorem indicates that prime numbers can be written as 6K-1 or 6K+1, with k a positive integer, but there are many K-values for both forms that do not generate prime numbers. This research focuses on the characteristics or patterns of these K-values that do not generate prime numbers for the two forms. The research shows that these values of K are infinite arithmetic progressions whose ratios are smaller prime numbers. This discovery allows the generalization of prime numbers through a sieve with the constraints of the values of K; and in turn, it is possible to prove the Goldbach Conjecture ("every even number greater than 2 can be written as the sum of two prime numbers") with simplicity of calculations and to deduce three corollaries from the same values of k are infinite arithmetic progressions whose ratios are smaller prime numbers. This discovery allows the generalization of prime numbers through a sieve with the constraints of the values of k; and in turn, it is possible to prove the Goldbach Conjecture ("every even number greater than 2 can be written as the sum of two prime numbers") with simplicity of calculations and to deduce three corollaries from the same.