Proof strategy based on factorization

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.

In order to prove the Theorem, first of all we build the factorization dashed line T of 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 connections between Bertrand’s postulate and Goldbach’s conjecture do not end with Goldbach-Bertrand’s theorem. Indeed, it can also be shown that Goldbach’s conjecture implies Bertrand’s postulate; that is, if Goldbach’s conjecture were true, it could be used to prove Bertrand’s postulate. More details can be found in the article Goldbach’s Conjecture Implies Bertrand’s Postulate by H. J. Ricardo. The proof is also shown on this page: https://proofwiki.org/wiki/Goldbach_implies_Bertrand.

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:

Figure 1: Proof strategy based on factorization. In the factorization dashed line of 2n=14, we must look for two spaces p and 2n-p (in green), both primes, whose sum is 2n. Existence of p and its primality are guaranteed by Bertrand’s postulate.

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.

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