Selberg’s sieve: statement and beggining of the proof
In this article we'll begin to get to the heart of sieve theory, analyzing in detail the so-called "Selberg's sieve".…
After almost three centuries, will we find a proof?
Category: Number theory
Why don’t algorithmic approaches work well in sieve theory? (part II)
In the previous article we calculated the sieve function of Erathostenes' sieve starting from the algorithm, obtaining a formula with…
Why don’t algorithmic approaches work well in sieve theory? (part I)
In the previous article we examined in detail the sieve of Eratosthenes, both at an algorithmic level and as a…
The sieve of Erathostenes and the formal definition of sieve
In the previous article we saw that the proof of Chen's Theorem is based on sieve theory. But what is…
Chen’s Theorem: statement and introduction to the proof
Chen's Theorem is one of the closest theorems most similar to Goldbach's Conjecture known so far. It is the work…
Our research
Currently, there are several attempts to prove the Goldbach's conjecture, which are complete, in the sense that they come to…
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…
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,…
The prime number Theorem: end of proof
In this post we'll complete the proof of the prime number Theorem, applying the fundamental ideas described in the previous…
The second part of the prime number Theorem proof: the basic ideas
In this post we’ll see what are the basic ideas of the second part of the prime number Theorem proof.…
Maximum space distance calculator
The Maximum space distance calculator is a program which we developed for computing the value of the maximum distance between…
End of the first part of the proof: the relationship between α and β’
With this post we'll conclude the main part of the Prime Number Theorem proof, which is based on the relationship…
A consequence of Selberg’s Theorem, in integral form
Looking back to the path travelled so far, we can identify a turning point: it was when we introduced Hypothesis…
Some important summations
When studying number theory, you'll soon realize that some familiarity with certain formalisms is required. In particular, some kinds of…
Selberg’s Theorem: proof and application
After the digression about the Möbius function of the last three posts, let's come back to the proof of the…
Factorizer
This page allows performing the decomposition of a number into its prime factors, and computing the value of some arithmetic…
Two lemmas with the Möbius function and the logarithm
In number theory, many proofs are "technical", i.e. they consist mainly in algebrical passages, by means of which an initial…
Other conjectures related to sums of primes
Over the centuries, various scholars have attempted to relate odd and even numbers with sums involving prime numbers. Some of…
The Möbius inversion formula
In the post Some important summations we introduced the summations extended to couples of variables the product of which divides…
The Möbius function and its connection with the function Λ
The properties of the divisors of natural numbers which we saw in the previous post let us define a function…
Bidimensional sieve of Eratosthenes
This page allows viewing a “bidimensional” version of the sieve of Eratosthenes applied to a given number. Differently from its…
Goldbach pairs viewer
This page allows viewing all Goldbach pairs in which an even number can be decomposed. The search can be done…
Two properties of the divisors of natural numbers
In this post we'll illustrate two properties of the divisors of natural numbers, starting from the simplest case, in which…
The integral mean value and the absolute error function R
In this post we'll apply the mean value Theorem for integrals in order to transform what we know about the…
Origins of the two Goldbach’s conjectures
Background Europe, 18th century. While the Western powers were all a flourishing of industries, cultural exchanges and scientific discoveries, the…
The functions W and V
The general idea of the Prime Number Theorem proof consists in starting from the proof of Chebyshev's Theorem (strong version),…
The Prime Number Theorem: history and statement
Looking at a prime numbers table, it's very simple to notice how their distribution seems to escape any regularity; instead…
Chebyshev’s Theorem (strong version)
In this post we'll revisit Chebyshev's Theorem, according to which the function π(x), that counts the number of prime numbers…
The factorial function and the Λ* function
Almost certainly you already know the factorial function, indicated by x!, which is read as "x factorial" and for an…
From integer numbers to real numbers – second part
In this post we'll see a technique that will let us overestimate or underestimate a value assumed by a function…
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…
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,…
Number theory
Goldbach's conjecture is put into the field of Number theory, the branch of Mathematics which studies the properties of integer…
Goldbach’s Conjecture
The proof of the Goldbach's Conjecture is one of the biggest still unsolved problems regarding prime numbers. Originally expressed in…
The sum of inverses of the first positive integers
In this post we'll analyze the sum of the first positive integers: 1 + 1/2 + 1/3 + 1/4 +…
From integer numbers to real numbers
So far we defined and studied only functions defined on integer numbers, the values of which can be integer or…
The lemma of bar chart area
The problem we establish in this post is to compute the area of a bar chart. Of course the area…
Chebyshev’s Theorem (weak version)
With this post we begin an analytical study of the function pi(x), that returns the number of primes less than…
The product of the first prime numbers: an underestimation
We saw that the product of the first prime numbers can be overestimated by a function of exponential kind with…
The least common multiple of the first positive integers
We know that a way to compute the least common multiple between two or more integer numbers is based on…
Bertrand’s postulate
The goal of this post is to prove the Bertrand's postulate, proposed in 1845 by the French mathematician Joseph Louis…
The product of the first prime numbers: an overestimation
A way to start investigating the sequence of prime numbers is to consider, starting from the beginning, portions of increasing…
Binomial estimates
Binomial coefficients are important for studying prime numbers. In this post we see in particular how to estimate, both upwards…
The definition of prime number
We'll start our study of prime numbers explaining the definition of prime number. It's commonly known that a prime number…
Some important results
The statement phrased by Christian Goldbach is a "conjecture", so, as a matter of principle, it is a hypothesis. This…
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