Brocard's conjecture
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, ... OEIS: 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 pn ≥ 3 since pn+1 − pn ≥ 2.
See also
- Prime-counting function
Notes
- ^ Weisstein, Eric W. "Brocard's Conjecture". MathWorld.
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Prime number conjectures
- Hardy–Littlewood
- 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
- Legendre's constant
- Lemoine's
- Mersenne
- Oppermann's
- Polignac's
- Pólya
- Schinzel's hypothesis H
- Waring's prime number
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