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 2024.
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
Notes