# 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.

## 6. The downcast problem

The difficulty of computing the t, t_value and t_space functions, increases as the order of the dashed line does. Their computation is not of particular difficulty for first order linear dashed lines, but things get more complicated when passing to higher order dashed lines. The downcast technique lets compute these functions for a dashed line of any order, by reducing to compute the same functions on one of its lower order dashed sublines.

## 7. Downcast characteristic equations of t and t_value for linear dashed lines

Solving the downcast problem, for linear dashed lines, means finding the solution of particular equations, called "downcast characteristic equations". In this post we'll see what equations let solve the downcast problem for the t and t_value functions.

## 8. Row computation in second order linear dashed lines

It may sound strange, but it's possible to compute a dash row without having an idea of its value (that is its column). In fact, row computation and value computation are two well distinct problems. In this post we'll se how to compute the row for second order linear dashed lines.

## 9. Row computation in third order linear dashed lines

For third order linear dashed lines, like second order ones, there exist some particular moduli which, by evaluating proper conditions, give us some information about the membership of a dash to a fixed row i. However, with respect to the second order, formulas are much more complex and, if you want to avoid annoying particular cases, you should consider dashed lines having two by two coprime components.

## 10. Row computation and differences between dash values

It's worth, before passing to the formulas of t_value, pausing for a while and looking closely to the moduli we introduced when analyzing row computation. In fact they have a very precise meaning related to the dashed line: they are directly connected with the difference between the value of a dash and the one of the previous dash of another row. In this post we'll see all that in detail, both for the second and the third order.

## 6. The downcast problem

The difficulty of computing the t, t_value and t_space functions, increases as the order of the dashed line does. Their computation is not of particular difficulty for first order linear dashed lines, but things get more complicated when passing to higher order dashed lines. The downcast technique lets compute these functions for a dashed line of any order, by reducing to compute the same functions on one of its lower order dashed sublines.

## 7. Downcast characteristic equations of t and t_value for linear dashed lines

Solving the downcast problem, for linear dashed lines, means finding the solution of particular equations, called "downcast characteristic equations". In this post we'll see what equations let solve the downcast problem for the t and t_value functions.

## 8. Row computation in second order linear dashed lines

It may sound strange, but it's possible to compute a dash row without having an idea of its value (that is its column). In fact, row computation and value computation are two well distinct problems. In this post we'll se how to compute the row for second order linear dashed lines.

## 9. Row computation in third order linear dashed lines

For third order linear dashed lines, like second order ones, there exist some particular moduli which, by evaluating proper conditions, give us some information about the membership of a dash to a fixed row i. However, with respect to the second order, formulas are much more complex and, if you want to avoid annoying particular cases, you should consider dashed lines having two by two coprime components.

## 10. Row computation and differences between dash values

It's worth, before passing to the formulas of t_value, pausing for a while and looking closely to the moduli we introduced when analyzing row computation. In fact they have a very precise meaning related to the dashed line: they are directly connected with the difference between the value of a dash and the one of the previous dash of another row. In this post we'll see all that in detail, both for the second and the third order.