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.
In the previous article we examined in detail the sieve of Eratosthenes, both at an algorithmic level and as a sieve function. We have seen that there is a strong connection between the two formulations, as the sieve function calculates how many elements remain in the set {2, 3, 4, ..., N} after having applied the algorithm. But so, to calculate the sieve function, why not start from the study of the algorithm from which it originates? In this article and the next one we'll see that this idea does not work in practice, and we'll understand why.
In the previous article we saw that the proof of Chen's Theorem is based on sieve theory. But what is a sieve? In this article we'll explain it based on the most famous example of a sieve, that of Eratosthenes. It allows you to answer a simple question: given an integer N ≥ 2, what are the prime numbers less than or equal to N?
Chen's Theorem is one of the closest theorems most similar to Goldbach's Conjecture known so far. It is the work of the Chinese mathematician Chen Jingrun, who published the proof in 1966. The best known form of the statement is the following: "Every sufficiently large even number is the sum either of two primes, or of a prime and a semiprime (i.e. a product of two prime numbers). However, this is only the abbreviated form of the Theorem: indeed Chen proved a more specific result, where he tries to estimate in how many ways an even number can be represented…
In this post we'll complete the proof of the prime number Theorem, applying the fundamental ideas described in the previous post. We'll split this last part of the proof into five parts: Our starting point will be a simple overestimation of the integral of |V| in any interval, obtained by applying Property A.18; We'll improve the result of the previous point supposing that within the interval the function V has at least one zero; We'll improve the result of the first point also for the intervals in which the function V has no zeroes; We'll compute how long the length…
In this post we’ll see what are the basic ideas of the second part of the prime number Theorem proof. Up to now we proved that α ≤ β′, where α and β′ are two important constants connected with the function V. As we saw, if we proved that α = 0 we would have proved the prime number Theorem. In order to prove that, a first idea may be proving that che β′ = 0, because it would follow that α ≤ β′ = 0, hence α = 0. In this post we’ll try to prove that β′ =…