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 $n={10}^4$.

Title	Brocard’s conjecture
Canonical name	BrocardsConjecture
Date of creation	2013-03-22 16:40:53
Last modified on	2013-03-22 16:40:53
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	8
Author	PrimeFan (13766)
Entry type	Conjecture
Classification	msc 11A41
Synonym	Brocard conjecture
Related topic	LegendresConjecture