Brocard's conjecture

In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (p_n)² and (p_n+1)², where p_n is the n^th prime number, for every n ≥ 2.^[1] The conjecture is named after Henri Brocard. It is widely believed that this conjecture is true^[2]. However, it remains unproven as of 2025. 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 p_n ≥ 3 since p_n+1 − p_n ≥ 2^[3].

Let ${\displaystyle p_{n}}$ be the ${\displaystyle n}$-th prime, and let ${\displaystyle \pi (x)}$ be the number of prime numbers ${\displaystyle \leq x}$. Formally, Brocard's conjecture claims:

${\displaystyle \pi {\big (}(p_{n+1})^{2}{\big )}-\pi {\big (}(p_{n})^{2}{\big )}\geq 4\quad {\text{for }}n\geq 2}$

This is equivalent to saying that there are at least ${\displaystyle 4}$ primes between squared consecutive primes other than ${\displaystyle 2}$ and ${\displaystyle 3}$.

Legendre's conjecture claims that there is a prime number between ${\displaystyle (n)^{2}}$ and ${\displaystyle (n+1)^{2}}$ for all natural number ${\displaystyle n}$. 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^[4]:

${\displaystyle \pi {\big (}(p_{n+1})^{2}{\big )}-\pi {\big (}(p_{n})^{2}{\big )}\geq 2\quad {\text{for }}n\geq 3}$

Cramér's conjecture claims that ${\displaystyle p_{n+1}-p_{n}=O((\log p_{n})^{2})}$, which gives a bound on how far apart primes can be. Cramér's conjecture implies Brocard's conjecture for sufficient ${\displaystyle n}$^[3].

Oppermann's conjecture claims that there is a prime in the interval ${\displaystyle (n,n(n+1))}$ and in the interval ${\displaystyle (n(n+1),(n+1)^{2})}$. This unsolved problem directly implies Brocard's conjecture.

n	${\displaystyle p_{n}}$	${\displaystyle p_{n}^{2}}$	Prime numbers	${\displaystyle \Delta }$
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
${\displaystyle \Delta }$ stands for ${\displaystyle \pi (p_{n+1}^{2})-\pi (p_{n}^{2})}$.

It is easy to verify the conjecture for small ${\displaystyle n}$:

${\displaystyle \pi {\big (}5^{2}{\big )}-\pi {\big (}3^{2}{\big )}=9-4\geq 4,\quad \pi {\big (}7^{2}{\big )}-\pi {\big (}5^{2}{\big )}=15-9\geq 4}$

The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... OEIS: A050216. See the table (right) for a list of primes sorted by the difference. See the animation (right) for the first ${\displaystyle 30}$ differences.

A trivial result from Bertrand's postulate, a proven theorem, states that because there is a prime in the interval ${\displaystyle (n,2n)}$, and the length of the interval ${\displaystyle ((p_{n})^{2},(p_{n+1})^{2})}$ is much greater than ${\displaystyle (p_{n},2p_{n})}$, Bertrand's postulate suggests many primes in the interval ${\displaystyle ((p_{n})^{2},(p_{n+1})^{2})}$, though not a sharp bound.

Using the bound proven by Baker et al.^[5], that ${\displaystyle p_{n+1}-p_{n}<p_{n}^{0.525}}$, one can show that there exist infinitely many ${\displaystyle p_{n}}$ such that there is at least one prime in the interval ${\displaystyle ((p_{n})^{2},(p_{n+1})^{2})}$, 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 ${\displaystyle n}$, 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 ${\displaystyle p_{n+1}-p_{n}=O{\big (}{\sqrt {p_{n}}}\log(p_{n}){\big )}}$, which implies Brocard's conjecture for sufficiently large ${\displaystyle n}$, similarly to Cramér's conjecture^[6].