  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 $n$.
