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).
See also
Andrica's Conjecture,
Cramér Conjecture,
Legendre's
Conjecture
Explore with Wolfram|Alpha
References
Brocard, H. Response to Problem 2181 in L'intermédiaire des math. 11, 149, 1904.Cohen, J. E. "Conjectures
about Primes and Cyclic Numbers." J. Int. Seq. 28, 25.4.7, 2025.
https://cs.uwaterloo.ca/journals/JIS/VOL28/Cohen/cohen41.pdf.Dickson,
L. E. History
of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York:
Dover, p. 436, 2005.Ribenboim, P. The
New Book of Prime Number Records. New York: Springer-Verlag, p. 248,
1996. https://doi.org/10.1007/978-1-4612-0759-7.Shannon,
A. G. and Leyendekkers, J. V. "On Legendre's Conjecture." Notes
Number Th. Disc. Math. 23, 117-125, 2017.Sloane, N. J. A.
Sequences A050216, A380135,
and A380136 in "The On-Line Encyclopedia
of Integer Sequences."Referenced on Wolfram|Alpha
Brocard's Conjecture
Cite this as:
Weisstein, Eric W. "Brocard's Conjecture."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BrocardsConjecture.html
Subject classifications
