Formulas giving prime numbers under Cram\'er's conjecture

Abstract

Under Cram\'er's conjecture concerning the prime numbers, we prove that for any $x>1$, there exists a real $A=A(x)>1$ for which the formula $[A^{n^x}]$ (where $[]$ denotes the integer part) gives a prime number for any positive integer $n$. Under the same conjecture, we also prove that for any $\epsilon>0$, there exists a positive real number $B$ for which the formula $[B.{n!}^{2+\epsilon}]$ gives a prime number for any sufficiently large positive integer $n$.