Function f(x) is the most closest prime number not less than $x$.

$f(5)=5\qquad f(9)=11$

Conjecture: $ x\leqslant f(x)\leqslant x+x^{\log_{113}13} \{x|1\leqslant x\leqslant+\infty,x\in \mathrm{positive~integer}\}$?

States that there is a prime number between $x$ and $x+x^{\log_{113}13}$ for every positive integer $x$?

For all positive integer $n$, where $p_n$ is the $n$th prime number.

$p_{n+1}-1-p_n\leqslant {\left(p_n\right)}^{\log_{113}13} \qquad$ ?