Currently, there are several attempts to prove the Goldbach’s conjecture, which are complete, in the sense that they come to the conclusion, but contain a series of problems, so they have not been considered valid by the international community. We instead propose some proof strategies, which are still far from being complete, but already contain interesting and unrefuted (so far) intermediate results. We hope this material will be a useful starting point for who, like us, has started looking for a proof.

We have tried to put our research in order, distinguishing between the proof strategies we use and the results achieved so far. Before going into detail, however, it is necessary to know that all our proof attempts are based on a theory called dashed line theory, which provides some tools for studying some relationships between integers, such as the divisibility relationship. To understand what this theory has to do with prime numbers, you can read the article From prime numbers to dashed lines.

## Our proof strategies

The concept of dashed line and the various concepts connected to it, which will be recalled at the appropriate moment, allow us to study Goldbach’s Conjecture with constructive type proof strategies, i.e. whose purpose is not only determining the existence of a solution, but also locating it, with a degree of accuracy that can vary depending on the strategy adopted. The details are explained in several articles, which are mutually linked. To make these links more apparent we have used a right pointing arrow icon, as in the following link, which leads to the initial article explaining the overall proof strategies:

Our proof strategies: an overview

Furthermore, a map is available showing which are the main articles (excluding the ones dedicated to the results that have been obtained so far, listed in the next paragraph) and the links between them:

Map of paths for proof strategies

During the reading you will find highlighted some points that are still open, which can be starting points for further insights. We are not able to go into all of them in depth, therefore we would very much like to receive contributions from outside. If you think about contributing, remember that we’re awarding prizes for who manages to solve some of these open points.< /p>

## The results achieved so far

The proof strategies listed above have led us to some partial results, both empirical (evidence based on computer simulations) and theoretical (real proofs). They can be fully understood only in the context of the proof strategies from which they originate, however we have also tried to introduce them independently, so that anyone can evaluate their scope on the basis of their mathematical knowledge.

Here is the list of results that we consider most significant:

- Study of existence of complementary space pairs based on second order dashed lines
- Maximum distance between spaces: empirical results
- (we have other interesting results, which we will publish here)

## There would still be so much to say…

What we have told above, both in relation to the proof strategies and to the results, constitutes a significant, but not exhaustive, part of our research work. We also had other ideas for studying Goldbach’s conjecture, but for one reason or another they didn’t work: because they were wrong or too abstract, or because it didn’t add up as we would have liked, or because we weren’t able to develop them sufficiently, or simply because we do not consider them worthy of note. The time available to us does not allow us to publish these ideas and any partial results connected to them; however, who is interested in furthering our research, even beyond what is published here, can contact us.