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

TL;DR

This paper presents a proof of Legendre's conjecture, demonstrating the existence of a prime between consecutive squares using a novel scheme involving 'active' sets and 'critical' elements.

Contribution

The paper introduces a new method for proving Legendre's conjecture by analyzing specific sets and elements associated with perfect squares.

Findings

Proof of Legendre's conjecture for all m ≥ 3

Introduction of 'active' sets and 'critical' elements in prime analysis

A new scheme for identifying primes between squares

Abstract

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 $m \ge 3$.