Is there a prime number between every prime and its square?
For each prime number $p$, is there always an other prime number between $p$ and $p^2$ ?
I tested it for prime numbers $< 500,000,000$, but I wanted to know if there is any mathematical proof of this ?
Yes. By Bertrand's postulate (actually a theorem), for every natural number $n$ (and thus every prime) there is a prime between $n$ and $2n$. As $p^2 \gt 2p$ for all primes $p$ greater than $2$, there is another prime in this interval, and when $p=2$, $3$ comes between $p$ and $p^2$.