# Brocard's conjecture

In number theory, Brocard's conjecture is a conjecture that there are at least four prime numbers between (pn)2 and (pn+1)2, for n > 1, where pn is the nth prime number.[1] It is widely believed that this conjecture is true. However, it remains unproven as of 2010.

n $p_n$ $p_n^2$ Prime numbers $\Delta$
1 2 4 5, 7 2
2 3 9 11, 13, 17, 19, 23 5
3 5 25 29, 31, 37, 41, 43, 47 6
4 7 49 53, 59, 61, 67, 71… 15
5 11 121 127, 131, 137, 139, 149… 9
$\Delta$ stands for $\pi(p_{n+1}^2) - \pi(p_n^2)$.

The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... (sequence A050216 in OEIS).

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 pn ≥ 3 since pn+1 - pn ≥ 2.