A conditional proof of Legendre's Conjecture and Andrica's conjecture

TL;DR

This paper proves that the Parity conjecture implies both Legendre's and Andrica's conjectures, offering a conditional proof that reduces the assumptions needed compared to previous approaches.

Contribution

It introduces the Parity conjecture and demonstrates that it implies two longstanding conjectures in number theory, providing a new conditional pathway for their proof.

Findings

Parity conjecture implies Legendre's conjecture

Parity conjecture implies Andrica's conjecture

Significant reduction in assumptions needed for proof

Abstract

The Legendre conjecture has resisted analysis over a century, even under assumption of the Riemann Hypothesis. We present, a significant improvement on previous results by greatly reducing the assumption to a more modest statement called the Parity conjecture. Let $p_n$ and $p_{n+1}$ be two consecutive odd primes, let $m$ be their midpoint fixed once for all. Conjecture: The largest multiple of $p_n$ not exceeding ${m_i}^2$ is odd for every integer $m_i$ in the interval $(p_n, m]$. Main result: We prove that the Parity conjecture implies Legendre's conjecture and Andrica's conjecture.