Boudaoud: La conjecture de Dickson et classes particulières d’entiers

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Abdelmadjid Boudaoud
La conjecture de Dickson et classes particulières d’entiers
Annales mathématiques Blaise Pascal, 13 no. 1 (2006), p. 103-109, doi: 10.5802/ambp.215
Article PDF | Reviews MR 2233013 | Zbl 1172.11307
Class. Math.: 11N32, 26E35, 11A51, 11A41, 11B83
Keywords: Conjecture de Dickson, analyse non standard, nombres premiers, suites d’entiers naturels.

Résumé - Abstract

As a consequence of Dickson’s Conjecture, we prove, for each couple of integers $q>0$ and $k>0$, the existence of an infinite set $L_{q,k}\subset \mathbb{N}$ such that, for each $n\in L_{q,k}$ and every integer $s$, $0<\left|s\right|\le q$, we have $n+s=\left|s\right|t_{1}...t_{k}$ where $t_{1}<...<t_{k} $are prime numbers. Similarly, we prove the existence of an infinite set $M_{q,k}\subset \mathbb{N}$ such that , for each $n\in M_{q,k}$ and every integer $s\in \left[ -q,q\right] $ (including $0$), we have $n+s=lt_{1}...t_{k}$ where $t_{1}<...<t_{k} $ are prime numbers and $l\in \left[ 1,2q+1\right] $ is an integer. The nonstandard interpretation of this result suggests the following question: Is every unlimited integer equal to the sum of a limited integer and a product of two unlimited integers ? We present families of integers in which each unlimited member is a product of two unlimited integers.

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