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

Apoloniusz Tyszka (1)

1 University of Agriculture in Krakow

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.

Mots clés: computation of length n; prime numbers of the form n!+1; prime numbers of the form n!-1; prime numbers p such that p!+1 is prime; twin prime conjecture

Domaines: Théorie des nombres [math.NT]; Logique [math.LO]

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-01614087v4⟩