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

Apoloniusz Tyszka (1)

1 Cracow University of Agriculture

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.

Mots clés: prime numbers of the form n!+1; equation y!=x(x+1)

Domaines: Théorie des nombres [math.NT]

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