In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (p<sub>n</sub>)<sup>2</sup> and (p<sub>n+1</sub>)<sup>2</sup>, where p<sub>n</sub> is the n<sup>th</sup> prime number, for every n âÂÂ¥ 2. The conjecture is named after Henri Brocard. It is widely believed that this conjecture is true. However, it remains unproven as of 2025. Legendre's conjecture, which states that there is a prime between consecutive integer squares, directly implies that there are at least two primes between prime squares for p<sub>n</sub> âÂÂ¥ 3 since p<sub>n+1</sub> â p<sub>n</sub> âÂÂ¥ 2.
Let be the -th prime, and let be the number of prime numbers . Formally, Brocard's conjecture claims:
This is equivalent to saying that there are at least primes between squared consecutive primes other than and .
Legendre's conjecture claims that there is a prime number between and for all natural number . It is an unsolved problem in mathematics as of 2025. If Legendre's conjecture is true, it immediately implies a weak version of Brocard's conjecture:
Cramér's conjecture claims that , which gives a bound on how far apart primes can be. Cramér's conjecture implies Brocard's conjecture for sufficient .
Oppermann's conjecture claims that there is a prime in the interval and in the interval . This unsolved problem directly implies Brocard's conjecture.
We begin with the fact that , meaning that the minimal interval between primes is . Then, according to Oppermann's conjecture, there is a prime in the interval , a prime in the interval , a prime in the interval , and a prime in the interval . Then, we have:
Which implies at least primes between and , and because , there are at least primes between any two squared consecutive primes, which is exactly what Brocard's conjecture claims.
It is easy to verify the conjecture for small :
The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... . See the table (right) for a list of primes sorted by the difference. See the animation (right) for the first differences.
A trivial result from Bertrand's postulate, a proven theorem, states that because there is a prime in the interval , and the length of the interval is much greater than , Bertrand's postulate suggests many primes in the interval , though not a sharp bound.
Using the bound proven by Baker et al., that , one can show that there exist infinitely many such that there is at least one prime in the interval , which is a much weaker result than Brocard's conjecture.
As shown above, Legendre's conjecture implies a weak version of Brocard's conjecture but is a strictly weaker conjecture.
As shown above, Oppermann's conjecture directly implies Brocard's conjecture for large enough , which constitutes a proof of Brocard's conjecture.
As shown above, Cramér's conjecture implies Brocard's conjecture directly.
The Riemann Hypothesis implies the bound , which implies Brocard's conjecture for sufficiently large , similarly to Cramér's conjecture.