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 (pn)2 and (pn+1)2, where pn is the nth 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 2022.

n Prime numbers
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
stands for .

The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... OEISA050216.

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+1pn ≥ 2.

See also edit

Notes edit

  1. ^ Weisstein, Eric W. "Brocard's Conjecture". MathWorld.