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...

- 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...

# Latest posts

In the previous post we introduced a new kind of summation, the one extended to the divisors of a positive integer number. Summations of this kind are often used in number theory, so it's worth to analyse them in more details. The most interesting case is the one of double summations extended to divisors: we'll see that they can be written in simpler ways, especially when the Möbius function comes into play.

The properties of the divisors of natural numbers which we saw in the previous post let us define a function which is very important in number theory, the Möbius function, indicated by the symbol μ. It's often used in summations in which the variable does not vary between a minimum and a maximum, but it assumes as possible values all and only the positive divisors of a natural number.

In this post we'll illustrate two properties of the divisors of natural numbers, starting from the simplest case, in which we'll take into consideration the numbers which are the product of two distinct prime factors. Let's take for example the number 10, which is the product of 2 and 5. So its unique non-trivial divisors, the ones different from the number itself and 1, are just 2 and 5. You can note that the number of trivial divisors (1 and 10) is equal to the number of non-trivial ones (2 and 5), and also their product is equal: 1 *…

In this post we'll apply the mean value Theorem for integrals in order to transform what we know about the integral of the function V into a knowledge about the values assumed by the function, aiming to prove the sufficient condition for the Prime Number Theorem, that is lim sup |V(u)| = 0. Our calculations will let us introduce a new function R that represents an absolute error, contrary to the functions V and W that represent relative errors.

Background Europe, 18th century. While the Western powers were all a flourishing of industries, cultural exchanges and scientific discoveries, the Russian Empire was always one step behind, with an industry that was little more than artisanship, a feudal economy, and a public education that wasn't even worthy of the name. But the new Tsar Peter the Great, after touring Europe in search of allies against the Turks who threatened the southern borders, was able to see this abysmal gap with his own eyes, and decided that the time had come for a turning point. He immediately started a great reform…