TITLE: Do you know Legendre's conjecture ? Has it been proved?
QUESTION [9 upvotes]: Legendre's conjecture: proposed by Adrien-Marie Legendre, states that there is a prime number between $n^2$ and $(n + 1)^2$ for every positive integer $n$.
Has it been proved?

REPLY [5 votes]: I realize this is an old post, however...
Legendre's conjecture is still open to date.  However, I recently completed my thesis, and as a corollary, show that there is always a prime between $n^2$ and $(n+1)^{2.000001}$... So close!
Of course one may also show $n^2 < p < (n+1)^{2+\varepsilon}$ for any $\varepsilon>0$.
I saw a few comments concerning Guerdes paper, which I emailed him about years ago.  Ultimately the paper is flawed.  It's been a few weeks since I saw it, however, so I will try my best to recall the error(s).
If referencing the 2013 paper, they say after (17) 
$\pi(2n)-\pi(n)+\text{Sum}_1-\text{Sum}_2,$
which is where the error lies.  You cannot make the subtraction summation 0 and still assume the inequality holds.  Consider $a+b+c-d<f$, but $a+b$ is not necessarily less than $f$ (let $a=4, b=2, c=1, d=3,$ and $f=5$).
Hope it helps.