$\hspace{10pt}$I was playing with R and a list of the first 10,000 primes and suddenly I had the idea of looking how many perfect squares are between $p_n$ and $p_{n+1}$.
$\hspace{10pt}$I make a program in R to compute it and I get an impressive result; we always have one or none. Also, I compute the number of times that we have a perfect square over the total gaps and my result was approximately $\frac{1}{3}$. In other words the probability that a perfect square is in $(p_{n},p_{n+1})$ is $\frac{1}{3}$.
Anyone here know some results related to this?