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.