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

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