The proof of the Goldbach's Conjecture is one of the biggest still unsolved problems regarding prime numbers. Originally expressed in 1742 by the mathematician Christian Goldbach, from whom the conjecture takes its name, it was rephrased by Euler in the form in which we know it today:
Every even number greater than 2 can be expressed as a sum of two prime numbers.
In spite of the empirical evidence and the simplicity of its statement, the conjecture is resisting to every attempts of proof since almost three centuries. Several mathematicians have proved some weaker versions of the conjecture.
In the previous article we tried to increase the number of spaces contained in a dashed line up to a certain column and we found that the error function that is obtained can assume very large values. However, this only happens when we assume that the final column can be any, while in our case we know that this column always belongs to the validity interval, which is a very small portion of the dashed line compared to the entire period. By introducing this constraint, the error function changes drastically...
In this formulation of the proof strategy based on spaces, we will look for conditions that allow us to conclude that a dashed line, single or double, has spaces in the interval (1, 2n), where 2n is the even number of the Conjecture. With a simple reasoning we conclude that it is sufficient that there exist at least two spaces in the first 2n columns of the dashed line, so we will look for a way to approximate the number of spaces contained up to a certain column, studying the error introduced by this approximation.
In the characterization of spaces we saw, for the first three orders of linear dashed lines, the formulas which calculate all and only the spaces that precede (or, respectively, that follow) a dash of component n_1 (the smallest component of the dashed line). In this article we'll take a more in-depth look at these formulas, arriving at giving a graphical interpretation.
Given an even number N > 2, we'll call Goldbach pairs for N the pairs of prime numbers the sum of which is N. Goldbach's Conjecture states that, whatever N, the number of these pairs is at least 1. In this article we'll reason in the opposite direction: we'll find an overestimate of the number of Goldbach pairs, that is, we'll establish how many they can be at most. We'll do this using the sieve theory developed so far, in particular Selberg's sieve.
This article concludes the "technical" part of the proof of Selberg's sieve and lays the foundation for the applications that we'll see later. The parts of the proof that we have seen in the previous articles will lead us, in a few steps, to the statement from which we started, Theorem C.4, which however is only a simplified version of the theorem. In this article we'll also state its ordinary form and we'll see that its proof differs only in a few points from that of the simplified version.