Abstract

Legendre's conjecture states that there is a prime number between $n^2$ and $(n+1)^2$ for every positive integer $n$.
In this paper we prove that every composite number between $n^2$ and $(n+1)^2$ can be written $u^2-v^2$ or $u^2-v^2+u-v$ that $u>0$ and $v\geq 0$. Using these result as well as induction and residues $(modq)$ we prove Legendre's conjecture.

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