## You are here

HomeBrocard's conjecture

## Primary tabs

# Brocard’s conjecture

(Henri Brocard) With the exception of 4 and 9, there are always at least four prime numbers between the square of a prime and the square of the next prime. To put it algebraically, given the $n$th prime $p_{n}$ (with $n>1$), the inequality $(\pi({p_{{n+1}}}^{2})-\pi({p_{n}}^{2}))>3$ is always true, where $\pi(x)$ is the prime counting function.

For example, between $2^{2}$ and $3^{2}$ there are only two primes: 5 and 7. But between $3^{2}$ and $5^{2}$ there are five primes: a prime quadruplet (11, 13, 17, 19) and 23.

This conjecture remains unproven as of 2007. Thanks to computers, brute force searches have shown that the conjecture holds true as high as $n=10^{4}$.

Related:

LegendresConjecture

Synonym:

Brocard conjecture

Type of Math Object:

Conjecture

Major Section:

Reference

## Mathematics Subject Classification

11A41*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections