Geometric representation of Goldbach pairs
When working with numbers, a geometric representation is often helpful in trying to highlight trends or properties that are less…
Multa non quia difficilia sunt non audemus sed quia non audemus sunt difficilia
Category: Dashed line theory
Overestimation of the number of spaces up to a column included into the validity interval
In the previous article we tried to increase the number of spaces contained in a dashed line up to a…
Proof strategy based on spaces: method based on the concept of maximum distance between consecutive spaces
Prerequisite: Proof strategy based on spaces As we stated in the previous post, our intent is to find the conditions…
Proof strategy based on spaces: method based on the approximate calculation of spaces
In this formulation of the proof strategy based on spaces, we will look for conditions that allow us to conclude…
21. Characterization of spaces: insights
In the characterization of spaces we saw, for the first three orders of linear dashed lines, the formulas which calculate…
Maximum distance between spaces: experimental results
Prerequisite: Proof strategy based on spaces One of the strategies that we have developed to try to prove Goldbach’s Conjecture…
Study about the existence of complementary space pairs based on second order dashed lines
Prerequisites: Dashed line theory definitions and symbols Our proof strategies Proof strategy based on dashes Characterization of spaces The aim…
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…
Proof strategy based on dashes
Prerequisite: Our proof strategies: an overview The proof strategy set out here starts from one of the assumptions of the…
Proof strategy based on spaces
Prerequisite: Our proof strategies: an overview The proof strategy which will be exposed here has the goal of proving Hypothesis…
Proof strategy based on factorization
Prerequisites: Our proof strategies Factorization dashed lines The final aim of the proof strategies that we are carrying out, as…
Our proof strategies: an overview
Prerequisite: Goldbach’s conjecture As already indicated, our ultimate goal is to use dashed line theory to prove Goldbach’s conjecture. Dashed…
Map of paths for proof strategies
The following map presents an overview of all paths related to proof strategies: the items listed below correspond to paths,…
Maximum space distance calculator
The Maximum space distance calculator is a program which we developed for computing the value of the maximum distance between…
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…
19. Calculation of t_value for dashed lines of arbitrary order
The function [latex]\mathrm{t\_value}[/latex], by definition, indicates which column of a dashed line a dash belongs to. For this reason, in…
15. Factorization dashed lines
When constructing a linear dashed line, the sequence of primes is usually used as a list of components, until it…
18. Calculation of t_space for dashed lines of arbitrary order
By now, the formula for calculation of [latex]\mathrm{t\_space}[/latex] for a linear dashed line of any order is not yet known,…
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…
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 [latex]T = (p_1, p_2,…
Dashed line theory
Dashed line theory is a new mathematical theory which studies the connection between the sequence of natural numbers and their…
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…
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…
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…
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…
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,…
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…
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…
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…
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.…
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…
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…
Dashed line viewer
The dashed line viewer lets you to show a dashed line at your choice, with the possibility to highlight spaces…
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…
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…