* The [prime-counting function](https://en.wikipedia.org/wiki/Prime-counting_function) is the function counting the number of prime numbers less than or equal to some real number $x$.  It is denoted by $\pi{(x)}$. Using my computer I found that *for any positive integer $X\leq 10^{9}$,* $$\pi{(X+\ln^2{X})}-\pi{(X-\ln^2{X})}>\ln{X}$$

*  **Question**: Does the result hold for all positive integers $X$?