Brocard's conjecture

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In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (p_n)² and (p_n+1)², where p_n is the n^th prime number, for every n ≥ 2.^[1] The conjecture is named after Henri Brocard. It is widely believed that this conjecture is true. However, it remains unproven as of 2024.

n	${\displaystyle p_{n}}$	${\displaystyle p_{n}^{2}}$	Prime numbers	${\displaystyle \Delta }$
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
${\displaystyle \Delta }$ stands for ${\displaystyle \pi (p_{n+1}^{2})-\pi (p_{n}^{2})}$.

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