Prerequisites:

Properties of the spaces of the factorization dashed lines can be used to set up a particular proof strategy, which can use not only the dashed line theory, but also the known theorems about prime numbers.

For example, as a starting point for the proof, it is possible to take advantage of the Bertrand’s postulate, according to which there is a prime number p between n + 1 and 2n. In fact, this prime number, due to the Property L.F.3 (Prime spaces on the right side of the factorization dashed line of an even number), is also a space; so on the opposite side of the dashed line, in a symmetrical position, there will in turn be another space, for the Property L.F.2 (Symmetry of factorization dashed lines).

We have thus obtained a pair of spaces, one of which is a prime number. Based on this the following result is proved, which we have called *Goldbach-Bertrand Theorem*, given that it has a formulation similar to Goldbach’s conjecture, and its proof uses the Bertrand’s postulate. However, this is a temporary name, because it is possible that in the future this Theorem will be replaced by some better result, which would deserve more of such an important name.

Goldbach-Bertrand Theorem

Every even number 2n \gt 2 can be written as the sum of two integers, one of which is prime and the other is coprime with 2n.

Due to Bertrand’s postulate, there is certainly a prime p, between n + 1 and 2n.

Since p is a prime number, due to the Property L.F.3 (Prime spaces on the right side of the factorization dashed line of an even number) it is also a space of T;

Since the dashed line T is symmetrical due to the Property L.F.2 (Symmetry of factorization dashed lines), also q := 2n - p is in turn another space;

Since q is a space, by definition it is not divisible by any of the components of the dashed line;

Hence, q is not divisible by any of the prime factors of 2n, i.e., by definition, q is coprime with 2n;

Furthermore, we have p + q = p + (2n - p) = 2n, that is p + q = 2n.

In summary, we have found two numbers p and q, of which p is prime, q is coprime with 2n, and their sum is 2n, which was the thesis to be proved.

The Goldbach-Bertrand Theorem is a starting point, but it is still far from the final goal. In order to arrive to the proper proof of the Goldbach conjecture, it will be necessary to find a condition that allows to choose p so that also 2n - p is prime, that is the column number which is in a symmetrical position with respect to the column p.

A possible starting point is asking ourselves if there is a rule on how our potential values of 2n - p are made, which we will call q, of which we know so far some characteristics:

- q \leq n;
- q is a space of the factorization dashed line of 2n;
- q it is certainly odd, because, being 2 a component of the dashed line, if it were even it would not be a space.

This proof strategy is represented in the following image:

The next step is to tighten the circle: Goldbach-Bertrand’s theorem assures us that there is at least one pair (p, q) in which p is prime and q is coprime with 2n. So, in order to arrive to the Goldbach conjecture, we must prove that:

- When the pair (p, q) is unique, q is prime;
- When many pairs (p, q), exist, at least one of them is such that q is prime.

Investigations about this are still ongoing, and have not yet reached a conclusion.