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Category: Number theory

The Möbius function and its connection with the function Λ

The Möbius function and its connection with the function Λ

Scritto il January 15, 2021January 15, 2021
Pubblicato inNumber theory
The properties of the divisors of natural numbers which we saw in the previous post let us define a function…
Bidimensional sieve of Eratosthenes

Bidimensional sieve of Eratosthenes

Scritto il December 10, 2020December 10, 2020
Pubblicato inGoldbach's conjecture
This page allows viewing a “bidimensional” version of the sieve of Eratosthenes applied to a given number. Differently from its…
Goldbach pairs viewer

Goldbach pairs viewer

Scritto il November 29, 2020December 10, 2020
Pubblicato inGoldbach's conjecture
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

Two properties of the divisors of natural numbers

Scritto il November 18, 2020December 29, 2020
Pubblicato inNumber theory
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

The integral mean value and the absolute error function R

Scritto il September 28, 2020December 22, 2020
Pubblicato inNumber theory
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

Origins of the two Goldbach’s conjectures

Scritto il September 7, 2020November 4, 2020
Pubblicato inFun facts , Goldbach's conjecture , Number theory
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 functions W and V

Scritto il August 22, 2020December 22, 2020
Pubblicato inNumber theory
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

The Prime Number Theorem: history and statement

Scritto il July 11, 2020December 22, 2020
Pubblicato inFun facts , Number theory
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)

Chebyshev’s Theorem (strong version)

Scritto il June 7, 2020December 22, 2020
Pubblicato inNumber theory
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 Lambda* function

The factorial function and the Lambda* function

Scritto il April 10, 2020December 22, 2020
Pubblicato inNumber theory
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

From integer numbers to real numbers – second part

Scritto il February 28, 2020December 22, 2020
Pubblicato inNumber theory
In this post we'll see a technique that will let us overestimate or underestimate a value assumed by a function…
Calculation of t_value for dashed lines of arbitrary order

Calculation of t_value for dashed lines of arbitrary order

Scritto il January 11, 2020December 17, 2020
Pubblicato inDashed line theory , Goldbach's conjecture
Prerequisites: Dashed line theory definitions and symbols Computation of a dash value in a linear third order dashed line First…
Proof strategy based on factorization

Proof strategy based on factorization

Scritto il January 2, 2020November 20, 2020
Pubblicato inDashed line theory , Goldbach's conjecture
Prerequisites: Our proof strategies Factorization dashed lines Properties of the spaces of the factorization dashed lines can be used to…
Study of existence of complementary space pairs based on second order dashed lines

Study of existence of complementary space pairs based on second order dashed lines

Scritto il December 30, 2019December 7, 2020
Pubblicato inDashed line theory , Goldbach's conjecture
Prerequisites: Dashed line theory definitions and symbols Our proof strategies Proof strategy based on dashes Characterization of spaces The purpose…
Upper bound for maximum distance between consecutive spaces

Upper bound for maximum distance between consecutive spaces

Scritto il December 29, 2019December 1, 2020
Pubblicato inDashed line theory , Goldbach's conjecture
Prerequisite: Characterization of spaces General aspects One of the still open problems of dashed line theory is, given a linear…
Proof strategy based on spaces

Proof strategy based on spaces

Scritto il December 29, 2019November 20, 2020
Pubblicato inDashed line theory , Goldbach's conjecture
Prerequisite: Our proof strategies The proof strategy shown here is aimed at proving the Hypothesis H.1 (Hypothesis of existence of…
Our proof strategies: an overview

Our proof strategies: an overview

Scritto il December 18, 2019November 20, 2020
Pubblicato inDashed line theory , Goldbach's conjecture
Prerequisites: Goldbach’s conjecture From prime numbers to dashed lines Periodicity and symmetry in linear dashed lines As already stated, our…
Proof strategy based on dashes

Proof strategy based on dashes

Scritto il December 17, 2019November 20, 2020
Pubblicato inDashed line theory , Goldbach's conjecture
Prerequisite: Our proof strategies The proof strategy which will be exposed here starts from one of the premises of Hypothesis…
Number theory

Number theory

Scritto il November 25, 2019November 27, 2019
Pubblicato inNumber theory
Goldbach's conjecture is put into the field of Number theory, the branch of Mathematics which studies the properties of integer…
Our proof strategies

Our proof strategies

Scritto il November 12, 2019December 7, 2020
Pubblicato inGoldbach's conjecture
Currently, there are several attempts to prove the Goldbach's conjecture, which are complete, in the sense that they come to…
Goldbach’s conjecture

Goldbach’s conjecture

Scritto il November 11, 2019December 2, 2020
Pubblicato inGoldbach'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

The sum of inverses of the first positive integers

Scritto il August 28, 2019December 22, 2020
Pubblicato inNumber theory
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

From integer numbers to real numbers

Scritto il May 19, 2019December 22, 2020
Pubblicato inNumber theory
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 lemma of bar chart area

Scritto il April 22, 2019December 22, 2020
Pubblicato inFun facts , Number theory
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)

Chebyshev’s Theorem (weak version)

Scritto il March 23, 2019December 22, 2020
Pubblicato inNumber theory
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

The product of the first prime numbers: an underestimation

Scritto il March 22, 2019December 22, 2020
Pubblicato inNumber theory
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

The least common multiple of the first positive integers

Scritto il January 25, 2019December 22, 2020
Pubblicato inNumber theory
We know that a way to compute the least common multiple between two or more integer numbers is based on…
Bertrand’s postulate

Bertrand’s postulate

Scritto il December 18, 2018December 22, 2020
Pubblicato inNumber theory
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

The product of the first prime numbers: an overestimation

Scritto il December 15, 2018December 22, 2020
Pubblicato inNumber theory
A way to start investigating the sequence of prime numbers is to consider, starting from the beginning, portions of increasing…
Binomial estimates

Binomial estimates

Scritto il December 14, 2018December 22, 2020
Pubblicato inNumber theory
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

The definition of prime number

Scritto il December 14, 2018December 22, 2020
Pubblicato inNumber theory
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

Some important results

Scritto il December 13, 2018March 30, 2020
Pubblicato inGoldbach's conjecture , Number theory
The statement phrased by Christian Goldbach is a "conjecture", so, as a matter of principle, it is a hypothesis. This…

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  • Home
  • Goldbach’s conjecture
    • Origins of the two Goldbach’s conjectures
    • Some important results
    • Our proof strategies
    • Our readers’ contributions
  • Number theory
    • Foundations of number theory
    • Bertrand’s Postulate
    • Chebyshev’s Theorem (weak version)
    • Chebyshev’s Theorem (strong version)
    • The Prime Number Theorem: the “elementary” proof
    • Complementary material
    • Fun facts
  • Dashed line theory
  • Info
    • About us
    • The project
    • Donations
    • Prizes
  • ItalianoItaliano