\[{{v}_{p}}(n!)=\sum\limits_{i=1}^{\infty }{\left\lfloor \frac{n}{{{p}^{i}}} \right\rfloor }=\sum\limits_{i=1}^{\left\lfloor {{\log }_{p}}n \right\rfloor }{\left\lfloor \frac{n}{{{p}^{i}}} \right\rfloor }\]

\[\lim \sum\limits_{p\text{ nguyên tố}}^{p\le n}{\frac{1}{p}}=+\infty \]

\[\prod\limits_{p\le n}{p}\le {{4}^{n}}\]

\[\lim \frac{{{\left( \pi (n) \right)}^{2}}}{n}=+\infty \]