A common approach to Brocard's problem, the problem of the infinitude of primes of the form n^2+1, and the twin prime problem

Apoloniusz Tyszka (1)

1 Cracow University of Agriculture

Let f(3)=4, and let f(n+1)=f(n)! for every integer n \geq 3. For an integer n \geq 3, let \Phi_n denote the following statement: if a system S \subseteq {x_i!=x_{i+1}: 1 \leq i \leq n-1} \cup {x_i \cdot x_j=x_{j+1}: 1 \leq i \leq j \leq n-1} has at most finitely many solutions in integers x_1,...,x_n greater than 1, then each such solution (x_1,...,x_n) satisfies x_1,...,x_n \leq f(n). We conjecture that the statements \Phi_3,... \Phi_{16} are true. We prove: (1) if the equation x!+1=y^2 has only finitely many solutions in positive integers, then the statement \Phi_6 implies that each such solution (x,y) belongs to the set {(4,5),(5,11),(7,71)}; (2) the statement \Phi_9 proves the implication: if there exists an integer x such that x^2+1 is prime and x^2+1>f(7), then there are infinitely many primes of the form n^2+1; (3) the statement \Phi_{16} proves the implication: if there exists a twin prime greater than f(14), then there are infinitely many twin primes.

Mots clés: Brocard's problem; Brocard-Ramanujan Diophantine equation; prime numbers of the form n^2+1; single query to the halting oracle; twin prime conjecture

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

Apoloniusz Tyszka. A common approach to Brocard's problem, the problem of the infinitude of primes of the form n^2+1, and the twin prime problem. 2017. ⟨hal-01614087v1⟩