Dashed line theory

Dashed line theory is a new mathematical theory which studies the connection between the sequence of natural numbers and their divisibility relationship. Typical problems are the computation of the n-th natural number divisible by at least one of k fixed numbers, or not divisible by any of them. Due to this nature, the theory is suited for studying prime numbers with a constructive approach, inspired to the sieve of Eratosthenes...

The birth of the theory was among the school desks, with the aim of creating a useful tool for the proof of the Goldbach's conjecture.
When the maths teacher of Simone, during the third year of high school, told him about the Goldbach’s conjecture, he felt the irresistible desire for understanding it deeply, in order to try to prove it. Some classmates followed his first reasoning, but their interest didn’t come from the problem itself, but from the one million dollar prize which at that time would have been won by whoever was able to prove it. For Simone the prize was important too, but not as much as the desire for knowing the mechanisms that conceal themselves under the simple statement of the conjecture. So his classmates soon lost interest in that matter, while he was busy, in the next years, in developing a new mathematical theory that he believed a useful tool for the proof. He named it dashed line theory.
But why conceiving a new theory, just for solving a specific problem? He thought that, if nobody was able to prove the conjecture yet, a reason could be the lack of the right mathematical tools. So it was worth the effort to try with a new approach, based on a new theory.

In 2010 the theory was published for the first time in a complete form, in Italian, in the website http://teoriadeitratteggi.webnode.it, but with a very formal language, which leaves little space to intuition. This aspect has been improved since 2018, with the birth of the Let's prove Goldbach! project, thanks to which the topics of dashed line theory have been explained in a more instructive way, with the help of images and examples. The result are the posts listed below.

Still as a part of the Let's prove Goldbach! project, the theory was further developed, when looking for the proof of the Goldbach's conjecture. Soon we'll gather most of these developments into a new section of this site.

As an internal reference to this section of the site, we created a list of the definitions and of the adopted symbols.

16. Characterization of spaces

One of the open problems of dashed line theory is "characterizing" spaces, i.e. looking for a criterion that tells us when a certain column of a dashed line is a space and when it is not. Certainly, a first characterization is the definition of space itself, but in our research we have noticed that it is sometimes useful to find alternative criteria.

17. Upper bound for maximum distance between consecutive spaces

One of the still open problems of dashed line theory is, given a linear dashed line T = (p_1, p_2, \ldots p_k) of order k, finding an expression that is able to represent an upper bound for the maximum distance between a space and the one immediately preceding it. That is, given two columns s_1 and s_2 of a dashed line, both spaces, such that s_1 \lt s_2, without other spaces between them, the purpose of this investigation is finding h such that s_2 - s_1 \leq h, for each possible pair of consecutive spaces (s_1, s_2) of the dashed…

18. Calculation of t_space for dashed lines of arbitrary order

By now, the formula for calculation of \mathrm{t\_space} for a linear dashed line of any order is not yet known, but, how we have seen in the section about dashed line theory, some partial results exist, which could bring, once extended, to a general formula.

19. Calculation of t_value for dashed lines of arbitrary order

The function \mathrm{t\_value}, by definition, indicates which column of a dashed line a dash belongs to. For this reason, in principle, calculating this function is quite easy, because the definition itself shows us how to do it.

20. Double dashed lines

Our research about Goldbach's Conjecture led us, at one point, to extend the definition of linear dashed line. As we know, a linear dashed line is a function represented by a table with k rows and infinite numbered columns starting from 1, where, in each row, a dash appears for each column whose number is a multiple of a given fixed positive integer (one of the components of the dashed line). Sometimes it is necessary to consider tables in which the dashes appear not only in correspondence with the multiples, but also in correspondence with the numbers that have a…