Proof of Legendre's Conjecture

TL;DR

This paper proves that for sufficiently large positive integers, there is always a prime between n^2 and (n+1)^2, advancing the understanding of prime distribution in quadratic intervals.

Contribution

It provides a proof confirming Legendre's conjecture holds for all sufficiently large n, using prime power counting techniques.

Findings

Existence of a prime between n^2 and (n+1)^2 for large n

Supports Legendre's conjecture asymptotically

Introduces a method based on prime power analysis

Abstract

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 $p\leq n$ such that $p^{o_p(n)}||n$.