Brocard's conjecture

Introduction

In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between ² and ², where p_n is the n^th prime number, for every n ≥ 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_n ≥ 3 since p_n+1 − p_n ≥ 2.

Mathematical statement

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.

Relation to other open problems in mathematics

Legendre's conjecture

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

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

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.

Examples

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 for a list of primes sorted by the difference. See the animation for the first differences.

Current research and results

Unconditional results

Bertrand's postulate

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.

Baker-Harman-Pintz 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.

Conditional results

Legendre's Conjecture - weak version of Brocard's conjecture

As shown above, Legendre's conjecture implies a weak version of Brocard's conjecture but is a strictly weaker conjecture.

Oppermann's Conjecture - full proof of Brocard's conjecture

As shown above, Oppermann's conjecture directly implies Brocard's conjecture for large enough, which constitutes a proof of Brocard's conjecture.

Cramér's Conjecture - full proof of Brocard's conjecture

As shown above, Cramér's conjecture implies Brocard's conjecture directly.

The Riemann Hypothesis - full proof of Brocard's conjecture

The Riemann Hypothesis implies the bound, which implies Brocard's conjecture for sufficiently large, similarly to Cramér's conjecture.