Brocard's conjecture

In number theory, Brocard's conjecture is a conjecture that there are at least four prime numbers between (p_n)² and (p_n+1)², for n > 1, where p_n is the n^th prime number. It is widely believed that this conjecture is true. However, it remains unproven as of 2016.

The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ...  A050216.

Legendre's conjecture that there is a prime between consecutive integer squares directly implies that there are at least two primes between prime squares for p_n ≥ 3 since p_n+1 - p_n ≥ 2.