Given a positive integer, let’s call the difference between two consecutive potential typical primality radii of a « Goldbach gap ». There are exactly such Goldbach gaps. Let’s denote by the number of distinct Goldbach gaps, and let’s write , the -th such Goldbach gap in the increasing order. Thus and . Moreover, let’s denote by the multiplicity of . One has and . Writing , one gets . Let’s define for ranging from to so that , and if . One has . Hence , and . Therefore .
Given a positive integer , let (respectively ) be the minimal (respectively maximal) distance between two consecutive potential typical primality radii of . Moreover, let and . One has (this inequality will be proved in the next article).
Using the inequality of arithmetic and geometric means, one can write the following inequality: . As , one gets . So that . Obviously , hence . As , one finally gets , where by we mean and where is the twin prime constant. Hence there exists such that . Therefore every large enough even integer is the sum of two primes.