The proof of the Goldbach's Conjecture is one of the biggest still unsolved problems regarding prime numbers. Originally expressed in 1742 by the mathematician Christian Goldbach, from whom the conjecture takes its name, it was rephrased by Euler in the form in which we know it today:
Every even number greater than 2 can be expressed as a sum of two prime numbers.
In spite of the empirical evidence and the simplicity of its statement, the conjecture is resisting to every attempts of proof since almost three centuries. Several mathematicians have proved some weaker versions of the conjecture.
The statement phrased by Christian Goldbach is a "conjecture", so, as a matter of principle, it is a hypothesis. This means that it can be:
- True, that is all even numbers greater than 2 can be expressed as a sum of two prime numbers;
- False, that is at least one even number greater than two exists, which cannot be written as a sum of two prime numbers.
Currently, there are several attempts to prove the Goldbach's conjecture, which are complete, in the sense that they come to the conclusion, but contain a series of problems, so they have not been considered valid by the international community. We instead propose some proof strategies, which are still far from being complete, but already contain interesting and unrefuted (so far) intermediate results. We hope this material will be a useful starting point for who, like us, has started looking for a proof...
The statement phrased by Christian Goldbach is a "conjecture", so, as a matter of principle, it is a hypothesis. This means that it can be:
- True, that is all even numbers greater than 2 can be expressed as a sum of two prime numbers;
- False, that is at least one even number greater than two exists, which cannot be written as a sum of two prime numbers.
Goldbach's conjecture is put into the field of Number theory, the branch of Mathematics which studies the properties of integer numbers. In order to understand the proofs of the results similar to the conjecture, and very likely also to prove the conjecture itself, a sound knowledge of number theory is required. But this kind of knowledge rarely is part of a mathematician's curriculum...
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...
Goldbach's conjecture is put into the field of Number theory, the branch of Mathematics which studies the properties of integer numbers. In order to understand the proofs of the results similar to the conjecture, and very likely also to prove the conjecture itself, a sound knowledge of number theory is required. But this kind of knowledge rarely is part of a mathematician's curriculum...
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...
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When working with numbers, a geometric representation is often helpful in trying to highlight trends or properties that are less likely to emerge from a purely numerical analysis; for this reason, we are looking for a way to represent Goldbach pairs on a Cartesian plane. One possible way is to apply the characterization of spaces, which for third order dashed lines allows each space of the dashed line to be associated with a point on the Cartesian plane.
In the previous article we tried to increase the number of spaces contained in a dashed line up to a certain column and we found that the error function that is obtained can assume very large values. However, this only happens when we assume that the final column can be any, while in our case we know that this column always belongs to the validity interval, which is a very small portion of the dashed line compared to the entire period. By introducing this constraint, the error function changes drastically...
In this formulation of the proof strategy based on spaces, we will look for conditions that allow us to conclude that a dashed line, single or double, has spaces in the interval (1, 2n), where 2n is the even number of the Conjecture. With a simple reasoning we conclude that it is sufficient that there exist at least two spaces in the first 2n columns of the dashed line, so we will look for a way to approximate the number of spaces contained up to a certain column, studying the error introduced by this approximation.
In the characterization of spaces we saw, for the first three orders of linear dashed lines, the formulas which calculate all and only the spaces that precede (or, respectively, that follow) a dash of component n_1 (the smallest component of the dashed line). In this article we'll take a more in-depth look at these formulas, arriving at giving a graphical interpretation.