Brocard's conjecture

Introduction

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. 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 $p_n$ be the $n$-th prime, and let $\pi (x)$ be the number of prime numbers $\le x$. Formally, Brocard's conjecture claims:

$\pi ((p_{n+1}{)}^2)-\pi ((p_n{)}^2)\ge 4\text{for }n\ge 2$

This is equivalent to saying that there are at least four primes between squared consecutive primes other than $2$ and $3$.

Relation to other open problems in mathematics

Legendre's conjecture

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

$\pi ((p_{n+1}{)}^2)-\pi ((p_n{)}^2)\ge 2\text{for }n\ge 3$

Cramér's conjecture

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

Oppermann's conjecture

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

Template:Collapse top We begin with the fact that $p_{n+1}-p_n\ge 2$, meaning that the minimal interval between primes is $(p_n,p_n+2)$. Then, according to Oppermann's conjecture, there is a prime in the interval $(p_n^2,p_n(p_n+1))$, a prime in the interval $(p_n(p_n+1),(p_n+1{)}^2)$, a prime in the interval $((p_n+1{)}^2,(p_n+1)(p_n+2))$, and a prime in the interval $((p_n+1)(p_n+2),(p_n+2{)}^2)$. Then, we have:

$\underset{\text{At least }1\text{ prime}}{\underbrace{(p_n^2,p_n(p_n+1))}},\underset{\text{At least }1\text{ prime}}{\underbrace{(p_n(p_n+1),(p_n+1{)}^2)}},\underset{\text{At least }1\text{ prime}}{\underbrace{((p_n+1{)}^2,(p_n+1)(p_n+2))}},\underset{\text{At least }1\text{ prime}}{\underbrace{((p_n+1)(p_n+2),(p_n+2{)}^2)}}$

Which implies at least $4$ primes between $p_n^2$ and $(p_n+2{)}^2$, and because $p_n+2\le p_{n+1}$, there are at least $4$ primes between any two squared consecutive primes, which is exactly what Brocard's conjecture claims. Template:Collapse bottom

Examples

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

The equation $\pi ((p_{n+1}{)}^2)-\pi ((p_n{)}^2)$ graphed up to $n=30$. The dotted line is the threshold that Brocard's conjecture claims to hold for all $n\ge 2$.

It is easy to verify the conjecture for small $n$:

$\pi \left(5^2\right)-\pi \left(3^2\right)=9-4\ge 4,\pi \left(7^2\right)-\pi \left(5^2\right)=15-9\ge 4$

The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... Template:OEIS2C. See the table (right) for a list of primes sorted by the difference. See the animation (right) for the first 30 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 $(n,2n)$, and the length of the interval $((p_n{)}^2,(p_{n+1}{)}^2)$ is much greater than $(p_n,2p_n)$, Bertrand's postulate suggests many primes in the interval $((p_n{)}^2,(p_{n+1}{)}^2)$, though not a sharp bound.

Baker-Harman-Pintz bound

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

See also

• Prime-counting function
• Legendre's conjecture
• Oppermann's conjecture
• Cramér's conjecture
• Riemann hypothesis
• Henri Brocard
• Big-O Notation
• Bertrand's postulate