[ Formalisation algébrique du crible d’ératosthène ]
Volume 77, Issue 1, February 2025, Pages 139–154
Mushiwalyahyage Zaluka1, Safari Mukeru2, and Déborah Amani Faraja3
1 Département de Mathématique-Physique, Institut Supérieur Pédagogique de Bukavu (ISP, BUKAVU), Bukavu, RD Congo
2 Department of Decision Sciences, School of Economics and Financial Sciences, College of Economics and Management Sciences, Université Sud-Africaine (UNISA), Pretoria, South Africa
3 Ecole des Mines, Université Officielle de Bukavu (U.O.B), Bukavu, RD Congo
Original language: French
Copyright © 2025 ISSR Journals. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper presents, in algebraic form, the set of prime numbers as obtained by the sieve of Eratosthenes and as contained in an arithmetic progression. In this way, it unifies old and recent studies on prime numbers: Euclid’s theorem, Dirichlet’s theorem, Green-Tao’s theorem, the conjecture of twin primes (generalized by Polignac) and Chebyshev’s Bias Phenomenon. It re-demonstrates the three theorems, solves Chebyshev’s Bias Phenomenon, demonstrates the twin primes conjecture and elucidates Polignac’s conjecture.
Author Keywords: number, composite, prime, twins, sieve, Eratosthenes, theorem, conjecture.
Volume 77, Issue 1, February 2025, Pages 139–154
Mushiwalyahyage Zaluka1, Safari Mukeru2, and Déborah Amani Faraja3
1 Département de Mathématique-Physique, Institut Supérieur Pédagogique de Bukavu (ISP, BUKAVU), Bukavu, RD Congo
2 Department of Decision Sciences, School of Economics and Financial Sciences, College of Economics and Management Sciences, Université Sud-Africaine (UNISA), Pretoria, South Africa
3 Ecole des Mines, Université Officielle de Bukavu (U.O.B), Bukavu, RD Congo
Original language: French
Copyright © 2025 ISSR Journals. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper presents, in algebraic form, the set of prime numbers as obtained by the sieve of Eratosthenes and as contained in an arithmetic progression. In this way, it unifies old and recent studies on prime numbers: Euclid’s theorem, Dirichlet’s theorem, Green-Tao’s theorem, the conjecture of twin primes (generalized by Polignac) and Chebyshev’s Bias Phenomenon. It re-demonstrates the three theorems, solves Chebyshev’s Bias Phenomenon, demonstrates the twin primes conjecture and elucidates Polignac’s conjecture.
Author Keywords: number, composite, prime, twins, sieve, Eratosthenes, theorem, conjecture.
Abstract: (french)
Cet article écrit, sous forme algébrique, l’ensemble des nombres premiers tels qu’ils s’obtiennent par le crible d’Eratosthène et tels qu’ils sont contenus dans une progression arithmétique. De cette manière, il unifie les études, anciennes et récentes, sur les nombres premiers: le théorème d’Euclide, le théorème de Dirichlet, le théorème de Green-Tao, la conjecture des nombres premiers jumeaux (généralisée par Polignac) et le Phénomène de Biais de Tchebychev. Il redémontre les trois théorèmes, résout le Phénomène de Biais de Tchebychev, démontre la conjecture des nombres premiers jumeaux et élucide la conjecture de Polignac.
Author Keywords: nombre, composé, premier, jumeaux, crible, Eratosthène, théorème, conjecture.
How to Cite this Article
Mushiwalyahyage Zaluka, Safari Mukeru, and Déborah Amani Faraja, “Algebraic formalization of the sieve of eratosthenes,” International Journal of Innovation and Scientific Research, vol. 77, no. 1, pp. 139–154, February 2025.
