Weak form of Brocard's conjecture I ask this out of curiosity, motivated by a question asked by one of my students.
The Brocard's conjecture claims that there exist at least four prime numbers between $p_{i}^2$ and $p_{i+1}^2$, where $p_i$ denotes the $i^{\rm th}$ prime number ($i>1$). 

Q. Is it known whether there exists at least one prime number between 
  $p_i^2$ and $p_{i+1}^2$?

 A: I am pretty sure that this is not known. If $p_{i+1}-p_i$ is bounded (which we know happens infinitely often), then $p_{i+1}^2-p_i^2\ll p_{i+1}\sim p_i$, so we are looking for primes in intervals essentially as short as in Legendre's conjecture. Of course, these intervals start at specific prime squares, but I don't see how this would make the problem any easier, i.e., how a very small prime gap at $p_i$ would help us finding another prime in a reasonably short interval starting at $p_i^2$. Just my two cents.
A: Not quite sure of what follows, but let's give it a try.
Suppose  $ n $ runs through a set of integers such that  $ (g_n) $  is not bounded.
Let  $ g_{n} : =p_{n+1}-p_{n} $, $ a_{i}: =\inf\{h\geq 0, g_{n}=O((\log_{i}(p_n))^h)\} $ where  $\log_{i}(n) $ is the  $ i $ -th iterate of the logarithm of  $ n $ and  $ j  : =\inf\{i\geq 0, a_{i}>0\} $ . Assume  $ j  $  is positive.
$ p^{2}_{n+1}-p^{2}_{n}=g_{n}(p_{n+1}+p_{n})>2g_{n}.p_{n}\gg 2p_{n}(\log_{j}p_n)^{a_{j}}\gg(\log_{j}(p^{2}_{n}))^{a_{j}+\varepsilon}$.
Hence  $ p^{2}_{n+1}-p^{2}_{n} $ is greater than the difference between the smallest prime greater than  $ p^{2}_{n} $ and  $ p^{2}_{n} $ , so there is a prime between  $p^{2}_{n} $ and  $ p^{2}_{n+1} $ .
