(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 th prime (with ), the inequality is always true, where is the prime counting function.

This conjecture remains unproven as of 2007. Thanks to computers, brute force searches have shown that the conjecture holds true as high as .