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Brocard's conjecture

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(Redirected from Brocard's Conjecture) Not to be confused with Brocard's problem.
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In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (pn) and (pn+1), where pn is the n prime number, for every n ≥ 2. The conjecture is named after Henri Brocard. It is widely believed that this conjecture is true. However, it remains unproven as of 2024.

n p n {\displaystyle p_{n}} p n 2 {\displaystyle p_{n}^{2}} Prime numbers Δ {\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
Δ {\displaystyle \Delta } stands for π ( p n + 1 2 ) π ( p n 2 ) {\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, ... 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

Notes

  1. Weisstein, Eric W. "Brocard's Conjecture". MathWorld.
Prime number conjectures


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