Proof of Legendre's Conjecture

Legendre’s conjecture states that there exists a prime between $n^2$ and $(n+1{)}^2$, for every positive integer $n$. Here I prove that for sufficiently large $n$, there is a prime number between $n^2$ and $(n+1{)}^2$. The proof relies on the idea of counting the maximum power, $o_p(n)$ of a prime $p\le n$ such that $p^{o_p(n)}\mathrm{\mid }\mathrm{\mid }n$.