Brocard's conjecture

Not to be confused with Brocard's problem.

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.^[1] It is widely believed that this conjecture is true. However, it remains unproven as of 2016.

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, ... 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.

Notes

↑ Weisstein, Eric W. "Brocard's Conjecture". MathWorld.

Prime number conjectures

Hardy–Littlewood 1st 2nd Agoh–Giuga Andrica's Artin's Bateman–Horn Brocard's Bunyakovsky Chinese hypothesis Cramér's Dickson's Elliott–Halberstam Firoozbakht's Gilbreath's Grimm's Landau's problems Goldbach's weak Legendre's Twin prime Legendre's constant Lemoine's Mersenne Oppermann's Polignac's Pólya Redmond–Sun Schinzel's hypothesis H Waring's prime number

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Brocard's conjecture

Notes

See also