# A common approach to the problem of the infinitude of twin primes, primes of the form n!+1, and primes of the form n!-1

Abstract : For a positive integer x, let \Gamma(x) denote (x-1)!. Let fact^{-1}:{1,2,6,24,...}-->N\{0} denote the inverse function to the factorial function. For positive integers x and y, let rem(x,y) denote the remainder from dividing x by y. For a positive integer n, by a computation of length n we understand any sequence of terms x_1,...,x_n such that x_1 is defined as the variable x, and for every integer i \in {2,...,n}, x_i is defined as \Gamma(x_{i-1}), or fact^{-1}(x_{i-1}), or rem(x_{i-1},x_{i-2}) (only if i \geq 3 and x_{i-1} is defined as \Gamma(x_{i-2})). Let f(4)=3, and let f(n+1)=f(n)! for every integer n \geq 4. For an integer n \geq 4, let \Psi_n denote the following statement: if a computation of length n returns positive integers x_1,...,x_n for at most finitely many positive integers x, then every such x does not exceed f(n). We prove: (1) the statement \Psi_4 equivalently expresses that there are infinitely many primes of the form n!+1; (2) the statement \Psi_6 implies that for infinitely many primes p the number p!+1 is prime; (3) the statement \Psi_6 implies that there are infinitely many primes of the form n!-1; (4) the statement \Psi_7 implies that there are infinitely many twin primes.
Keywords :
Type de document :
Pré-publication, Document de travail
2018
Domaine :
Liste complète des métadonnées

Littérature citée [8 références]

https://hal.archives-ouvertes.fr/hal-01614087
Contributeur : Apoloniusz Tyszka <>
Soumis le : samedi 24 mars 2018 - 03:25:08
Dernière modification le : mardi 27 mars 2018 - 01:03:43
Document(s) archivé(s) le : jeudi 13 septembre 2018 - 06:12:03

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• HAL Id : hal-01614087, version 5

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Apoloniusz Tyszka. A common approach to the problem of the infinitude of twin primes, primes of the form n!+1, and primes of the form n!-1. 2018. 〈hal-01614087v5〉

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