Do you know Legendre's conjecture ? Has it been proved? 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?
 A: The conjecture has not been proved. A weaker form of the conjecture would say that for any $x\ge0$ (not just integers), there is a prime between $x^2$ and $(x+2)^2$. Equivalently, the gap between $g_n=p_{n+1}-p_n$ is at most $4\sqrt{p_n}+4$. An even weaker conjecture is that there is some finite $k$ such that $g_n<k\sqrt{p_n}.$ But even this conjecture is not implied by the Riemann Hypothesis.
To be blunt: If Legendre's conjecture had actually been proved, the authors would not have to pay $100 to have the result published in a third-rate journal. You may wish to look at
http://www.scottaaronson.com/blog/?p=304
A: 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.
