Brocard's conjecture (Brocard 1904) states that
 |
for integers 
, where 
is the prime counting
function and 
is the 
th
prime.
The following table lists the primes between successive values of 
.
 |  |  | primes between  and  | number of primes |
| 1 | 2 | 3 | 5, 7 | 2 |
| 2 | 3 | 5 | 11, 13, 17, 19, 23 | 5 |
| 3 | 5 | 7 | 29, 31, 37, 41, 43, 47 | 6 |
| 4 | 7 | 11 | 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113 | 15 |
| 5 | 11 | 13 | 127, 131, 137, 139, 149, 151, 157, 163, 167 | 9 |
For 
,
2, ..., the first few values are 2, 5, 6, 15, 9, 22, 11, 27, 47, 16, ... (OEIS A050216), illustrated above.
The high-water marks for these values are 2, 5, 6, 15, 22, 27, 47, 57, 80, 90, 106, ... (OEIS A380135),
which occur for 
,
2, 3, 4, 6, 8, 9, 11, 15, 18, 21, ... (OEIS A380136).
When plotted as a scatter diagram, a sequence of rays radiating from the origin becomes evident.
The conjecture is true for 
(E. Weisstein, Jan. 12, 2025)
but remains unproven (Cohen 2025, p. 15).
