A conjecture which implies that any twin prime greater than ((((((((((((24!)!)!)!)!)!)!)!)!)!)!)!)!+3 guarantees that the set of twin primes is infinite
Résumé
Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n \geq 2. For a positive integer n, let \Gamma_n denote the following statement: if a system S \subseteq {x_i!=x_k: i,k \in {1,...,n}} \cup {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has at most finitely many solutions in integers x_1,...,x_n greater than 1, then each such solution (x_1,...,x_n) satisfies min(x_1,...,x_n) \leq f(n). We conjecture that the statements \Gamma_1,...,\Gamma_{16} are true. We prove: (1) if the equation x!+1=y^2 has only finitely many solutions in positive integers, then the statement \Gamma_6 guarantees that each such solution (x,y) satisfies x \leqslant f(6); (2) the statement \Gamma_9 proves the implication: if there exists an integer x>f(9) such that x^2+1 is prime, then there are infinitely many primes of the form n^2+1; (3) the statement \Gamma_{16} proves the implication: if there exists a twin prime greater than f(16)+3, then there are infinitely many twin primes.
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