A conjecture which implies that there are infinitely many primes of the form n!+1

Abstract : Let f(6)=720, and let f(n+1)=f(n)! for every integer n \geq 6. For an integer n \geq 6, let \Lambda_n denote the following statement: if a system S \subseteq {x_i!=x_j: 1 \leq i < j \leq n} \cup {x_i \cdot x_j=x_{j+1}: 1 \leq i < j \leq n-1} has at most finitely many solutions in integers x_1,...,x_n greater than 3, then each such solution (x_1,...,x_n) satisfies x_1,...,x_n \leq f(n). We conjecture that the statements \Lambda_6,...,\Lambda_9 are true. We prove that the statement \Lambda_9 implies that there are infinitely many primes of the form n!+1.
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Type de document :
Pré-publication, Document de travail
2017

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https://hal.archives-ouvertes.fr/hal-01625653
Contributeur : Apoloniusz Tyszka <>
Soumis le : lundi 13 novembre 2017 - 03:27:28
Dernière modification le : mardi 14 novembre 2017 - 01:02:40
Document(s) archivé(s) le : mercredi 14 février 2018 - 12:37:55

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erdos_problem.pdf
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• HAL Id : hal-01625653, version 2

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Apoloniusz Tyszka. A conjecture which implies that there are infinitely many primes of the form n!+1. 2017. 〈hal-01625653v2〉

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