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Under Cram\'er's conjecture concerning the prime numbers, we prove that forany $x>1$, 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 positiveinteger $n$. 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 largepositive integer $n$.

Formulas giving prime numbers under Cramér’s conjecture