Legendre's conjecture. Theorem on existence of a prime number between $m^{2}$ and $(m+1)^{2}$

In scientific paper, we will show a proof of Legendre's conjecture based on a scheme for finding elements of "active" set ${\rm H}_{m^{4}}$ and "critical" element $C_{m^{4}}$ for number $m^{4}$ at each $m \ge 3$.