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Brocard’s Conjecture: At Least Four Primes Between Consecutive Prime Squares

Establish whether, for every n ≥ 2 where p_n denotes the nth prime number, the open interval between p_n^2 and p_{n+1}^2 contains at least four primes.

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Background

Brocard’s conjecture concerns the density of primes between the squares of consecutive primes, positing a minimum of four primes in each such interval. It provides a strong criterion for prime distribution in short intervals.

The paper lists Brocard’s conjecture among foundational open problems relating to prime gaps, framing subsequent results about intervals and gap bounds.

References

Besides, Brocard conjectured that if $p_n$ is the $n$th prime with $n\ge2$, than there are at least four primes between $p_n2$ and $p_{n+1}2$; similarly, Andrica(1986) conjectured that if $p_n$ is the $n$th prime, then $\sqrt{p_{n+1}-\sqrt{p_n} < 1$.

— On the Maximal Gap between Primes (2510.17065 - Wang, 20 Oct 2025) in Section 1 (Introduction)