brocard

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. It is widely believed that this conjecture is true. However, it remains unproven as of 2007.

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

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.

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