Pré-Publication, Document De Travail Année : 2019

EVERY SUFFICIENTLY LARGE EVEN NUMBER IS THE SUM OF TWO PRIMES

Résumé

The binary Goldbach conjecture asserts that every even integer greater than $4$ is the sum of two primes. In this paper, we prove that there exists an integer $K_\alpha$ such that every even integer $x > p_k^2$ can be expressed as the sum of two primes, where $p_k$ is the $k$th prime number and $k > K_\alpha$. To prove this statement, we begin by introducing a type of double sieve of Eratosthenes as follows. Given a positive even integer $x > 4$, we sift from $[1, x]$ all those elements that are congruents to $0$ modulo $p$ or congruents to $x$ modulo $p$, where $p$ is a prime less than $\sqrt{x}$. Therefore, any integer in the interval $[\sqrt{x}, x]$ that remains unsifted is a prime $q$ for which either $x-q = 1$ or $x-q$ is also a prime. Then, we introduce a new way of formulating a sieve, which we call the sequence of $k$-tuples of remainders. By means of this tool, we prove that there exists an integer $K_\alpha > 5$ such that $p_k / 2$ is a lower bound for the sifting function of this sieve, for every even number $x$ that satisfies $p_k^2 < x < p_{k+1}^2$, where $k > K_\alpha$, which implies that $x > p_k^2 \; (k > K_\alpha)$ can be expressed as the sum of two primes.

Mots clés

Fichier principal
Vignette du fichier
RGBmanuscript191025.pdf (467.92 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Licence
Loading...

Dates et versions

hal-02075531 , version 1 (21-03-2019)
hal-02075531 , version 2 (24-07-2019)
hal-02075531 , version 3 (26-10-2019)
hal-02075531 , version 4 (01-08-2020)
hal-02075531 , version 5 (09-01-2021)
hal-02075531 , version 6 (22-03-2022)
hal-02075531 , version 7 (03-04-2024)

Licence

Identifiants

  • HAL Id : hal-02075531 , version 3

Citer

Ricardo Guillermo Barca. EVERY SUFFICIENTLY LARGE EVEN NUMBER IS THE SUM OF TWO PRIMES. 2019. ⟨hal-02075531v3⟩
2644 Consultations
1515 Téléchargements

Partager

  • More