Formulas giving prime numbers under Cramér's conjecture

Abstract

Under Cramér'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 $n$.

9 pages