Counting primes in the interval (n^2,(n+1)^2)

Abstract

In this note, we show that there are many infinity positive integer values of $n$ in which, the following inequality holds $$\lfloor 1/2(\frac{(n+1{)}^2}{\log (n+1)}-\frac{n^2}{\log n})-\frac{{\log }^{2}n}{\log \log n}\rfloor \le \pi ((n+1{)}^2)-\pi (n^2).$$

This is a three pages unsuccessful (but maybe useful) challenge, for proving the old-famous conjecture, which asserts for every positive integer n, the interval (n^2,(n+1)^2) contains at least a prime