Brocard's Conjecture (Prime Gaps)

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $p_n$ be the $n$th prime in increasing order, so $p_1=2$. Write $\pi(x)=\#\{p\leq x:p\text{ is prime}\}$. The standard Brocard assertion excludes $n=1$, as the catalogue background specifies: $(p_1^2,p_2^2)=(4,9)$ contains exactly the two primes $5$ and $7$. Endpoints $p_n^2$ are not prime.

Statement recorded in the supplied source.

For every integer $n\geq2$, are there at least four primes in the open interval $(p_n^2,p_{n+1}^2)$? Equivalently, is \[ \pi(p_{n+1}^2)-\pi(p_n^2)\geq4\qquad(n\geq2)? \] [S2]

Short English statement

Beyond the first pair of primes, do squares of consecutive primes always enclose at least four primes?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1294 (NT-086). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Germán Andrés Paz, On Legendre's, Brocard's, Andrica's, and Oppermann's Conjectures, arXiv:1310.1323v2 (2014), Section 3. https://arxiv.org/pdf/1310.1323.