Brocard's conjecture

{{Distinguish|Brocard's problem}}

{{one source|date=September 2015}}

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.{{mathworld|urlname=BrocardsConjecture|title=Brocard's Conjecture}} The conjecture is named after Henri Brocard. It is widely believed that this conjecture is true. However, it remains unproven as of 2025.

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n	$p\_n$	$p\_n^2$	Prime numbers	$\backslash Delta$
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
colspan=5| $\backslash Delta$ stands for $\backslash pi(p\_\{n+1\}^2)\; -\; \backslash pi(p\_n^2)$.

The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... {{OEIS2C|id=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.