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.

1. From prime numbers to dashed lines

A dashed line can be represented as a table having a fixed number of rows, and, potentially, infinite columns, numbered starting from zero. Table cells can be empty, or can contain a dash (-). If in every row, dashes are arranged starting from column 0 at regular intervals, the resulting configuration may be taken as a starting point for studying prime numbers...

2. Dashed lines, dashes and spaces: some definitions and simple properties

In the formal definition of dashed line, a dash is a couple of natural numbers (i, n), where i identifies the row in which the dash is placed, and n is the ordinal number that specifies its position inside the row. A dashed line is a function mapping each dash to its value, that is the column it belongs to. The columns which don't contain any dash are called spaces.

3. Periodicity and symmetry in linear dashed lines

In this post we'll talk about two fundamental properties of linear dashed lines: periodicity and symmetry. When we talk about periodicity we mean that in the graphical representation of a dashed line there are some outlines that recur infinitely many times, moreover some of them are symmetrical, in the sense that the first column is equal to the last one, the second column is equal to the second to last, etcetera.

4. From a problem about jogging to dashed lines

Charles, Mary and Tom are three friends who like jogging. Today they went jogging together, but each of them follow an own pace: Charles goes jogging every three days, Mary every four days and Tom every five. Given a positive integer x, who among them will be the x-th to go jogging, and in how many days he or she will do it? What will be the x-th day in which no one of them go jogging? The dashed line theory lets us answer this kind of questions. Thus, the definitions of the t, t_value and t_space functions have their…

5. First order linear dashed lines

We saw that one of the main problems of dashed line theory is computing the t, t_value and t_space functions. Now we'll see how to make this calculation for linear first order dashed lines, whereas the next posts will be devoted to higher order ones.

1. From prime numbers to dashed lines

A dashed line can be represented as a table having a fixed number of rows, and, potentially, infinite columns, numbered starting from zero. Table cells can be empty, or can contain a dash (-). If in every row, dashes are arranged starting from column 0 at regular intervals, the resulting configuration may be taken as a starting point for studying prime numbers...

2. Dashed lines, dashes and spaces: some definitions and simple properties

In the formal definition of dashed line, a dash is a couple of natural numbers (i, n), where i identifies the row in which the dash is placed, and n is the ordinal number that specifies its position inside the row. A dashed line is a function mapping each dash to its value, that is the column it belongs to. The columns which don't contain any dash are called spaces.

3. Periodicity and symmetry in linear dashed lines

In this post we'll talk about two fundamental properties of linear dashed lines: periodicity and symmetry. When we talk about periodicity we mean that in the graphical representation of a dashed line there are some outlines that recur infinitely many times, moreover some of them are symmetrical, in the sense that the first column is equal to the last one, the second column is equal to the second to last, etcetera.

4. From a problem about jogging to dashed lines

Charles, Mary and Tom are three friends who like jogging. Today they went jogging together, but each of them follow an own pace: Charles goes jogging every three days, Mary every four days and Tom every five. Given a positive integer x, who among them will be the x-th to go jogging, and in how many days he or she will do it? What will be the x-th day in which no one of them go jogging? The dashed line theory lets us answer this kind of questions. Thus, the definitions of the t, t_value and t_space functions have their…

5. First order linear dashed lines

We saw that one of the main problems of dashed line theory is computing the t, t_value and t_space functions. Now we'll see how to make this calculation for linear first order dashed lines, whereas the next posts will be devoted to higher order ones.