## On Legendre's, Brocard's, Andrica's, and Oppermann's Conjectures

**Authors:** Germán Paz

Let $n\in\mathbb{Z}^+$. Is it true that every sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1)^2$ contains at least one prime number? In this paper we show that this is actually the case for every $n \leq 1,193,806,023$. In addition, we prove that a positive answer to the previous question for all $n$ would imply Legendre's, Brocard's, Andrica's, and Oppermann's conjectures, as well as the assumption that for every $n$ there is always a prime number in the interval $[n,n+2\lfloor\sqrt{n}\rfloor-1]$.

**Comments:** 11 Pages. The title and the abstract have been modified; a few references have been added, as it has been suggested to the author. This paper is also available at arxiv.org/abs/1310.1323.

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### Submission history

[v1] 2012-08-06 18:11:53

[v2] 2013-08-11 17:54:23

[v3] 2013-09-06 11:35:40

[v4] 2014-04-03 22:09:17

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