This article is meant to be the last one on this blog, and aims at summing up the ideas developped in the previous articles to give a rather accurate sketch of proof of the asymptotic Goldbach conjecture.

First, let’s give all the relevant notations:

: a positive integer strictly greater than .

: a primality radius of , namely a non-negative integer less than such that both and are primes.

: the -tuple , where denotes the -th prime.

: where is the number of primes below .

: the product of the first primes.

: the -th potential typical primality radius of , i.e the -th non-negative integer less than such that both and contain no :

for all prime less than , one has simultaneously:

and

The adjective « typical » means that:

and the adjective « potential » is used because of the upper limit

A potential typical primality radius of less than will be simply called a typical primality radius of .

: the total number of potential typical primality radii of , an expression of which is:

.

: the total number of typical primality radii of .

: the quantity .

: the quantity

if

The quantities for ranging from to will be called « Goldbach gaps of the first kind ».

: a quantity such that:

if < then <

and such that there exists such that

The quantities will be called « Goldbach gaps of the second kind » and the number of such that:

that is, the multiplicity of , will be denoted by .

: the number of Goldbach gaps of the second kind.

: the ratio .

: the quantity equal to:

if and equal to otherwise.

: the quantity .

: the ratio .

The famous Goldbach conjecture asserts that every even integer greater than is the sum of two primes. We define the number , which depends on , in the following way:

where is the number of primes less or equal to .

is a prime only if for all prime less or equal to , doesn’t divide .

There are exactly such primes. The number will be called the « natural configuration order » of .

Then we define the « -order configuration » of an integer , denoted by , as the following sequence:

.

For example .

We call the « natural configuration » of .

An almost sufficient condition to make be a primality radius of is:

For all integer such that :

differs from

and

differs from

If this double statement is true, will be called a « potential typical primality radius » of .

Moreover, if , then will be called a « typical primality radius » of and denoted simply by .

In what follows we show that every large enough positive integer admits a typical primality radius, from which it follows that every large enough even integer is the sum of two primes.

The proof is based on two lemmas:

Lemma 1:

Proof:

One has:

But:

.

Using Mertens’ formula, namely:

one gets:

and thus:

where

is the so called twin prime constant.

Finally:

hence:

.

Lemma 2: Assume that .

Then .

Proof:

One has:

and:

thus:

Moreover, one has:

so that:

hence:

Hence:

and:

Therefore

Theorem:

Proof:

One can use the inequality of the arithmetic and geometric mean to write:

As , one gets:

As obviously and , one finally gets:

Hence .

All that remains to be done is to prove the inequality , which should be true whenever . This will be the subject of a future article.