From Wikipedia, the free encyclopedia
Not to be confused with Brocard's problem.
This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Brocard's conjecture" – news · newspapers · books · scholar · JSTOR (September 2015) |
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 2025.
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 |
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
[edit]Notes
[edit]
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