Number Theory


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

Download: PDF

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

Unique-IP document downloads: 562 times is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus