# 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.

## 11. Computation of a dash value in a linear second order dashed line

In this post we'll see how it's possible to compute the x-th dash value in a second order linear dashed line. For doing this, first the dash row is computed; then the dash ordinal inside that row has to be computed, solving the downcast characteristic equation; finally, having reduced the problem to the first order, the first order t_value formula can be used.

## 12. Computation of a dash value in a linear third order dashed line

In order to compute the x-th dash value in a third order linear dashed line, we can proceed in a similar manner as in the second order. But now we have two possibilities: we can do the downcast from the third order to the second one (that is from T to T[i, j], where T is the given dashed line), or directly from the third order to the first one (that is from T to T[i]).

## 13. Computation of spaces in second order linear dashed lines

In this post we'll see how to obtain a formula for computing the t_space function for second order linear dashed lines. Like we did for t and t_value functions, we'll follow the downcast approach, which is simpler in the case of t_space, because of its down-conservativity. We'll see first an approximated formula, which will be later modified for obtaining the exact one.

## 14. A recursive property of linear dashed lines with two by two coprime components

In this post we'll talk about a recursive property, that is true for all linear dashed lines with two by two coprime components. Given a dashed line T of that kind, we'll see that every its proper dashed subline T' can be found "immersed" into T; moreover, recursively, also every proper dashed subline T'' of T' can be found "immersed" into T', and so on.

## 15. Factorization dashed lines

When constructing a linear dashed line, the sequence of primes is usually used as a list of components, until it stops at one of them, according to a certain criterion. However, this is not a strict rule of dashed line theory, which, in principle, allows us to choose components arbitrarily. Obviously, the way to choose the components is important, because, depending on which and how many are used, the structure of the dashed line changes accordingly, and therefore its properties also change.

## 11. Computation of a dash value in a linear second order dashed line

In this post we'll see how it's possible to compute the x-th dash value in a second order linear dashed line. For doing this, first the dash row is computed; then the dash ordinal inside that row has to be computed, solving the downcast characteristic equation; finally, having reduced the problem to the first order, the first order t_value formula can be used.

## 12. Computation of a dash value in a linear third order dashed line

In order to compute the x-th dash value in a third order linear dashed line, we can proceed in a similar manner as in the second order. But now we have two possibilities: we can do the downcast from the third order to the second one (that is from T to T[i, j], where T is the given dashed line), or directly from the third order to the first one (that is from T to T[i]).

## 13. Computation of spaces in second order linear dashed lines

In this post we'll see how to obtain a formula for computing the t_space function for second order linear dashed lines. Like we did for t and t_value functions, we'll follow the downcast approach, which is simpler in the case of t_space, because of its down-conservativity. We'll see first an approximated formula, which will be later modified for obtaining the exact one.

## 14. A recursive property of linear dashed lines with two by two coprime components

In this post we'll talk about a recursive property, that is true for all linear dashed lines with two by two coprime components. Given a dashed line T of that kind, we'll see that every its proper dashed subline T' can be found "immersed" into T; moreover, recursively, also every proper dashed subline T'' of T' can be found "immersed" into T', and so on.