Is there a prime between n^(2)and (n+1)^(2 )for every n > 0

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.

A prime is an integer $p\ge2$ whose only positive divisors are $1$ and $p$. The interval between consecutive squares is taken to be open.

Statement recorded in the supplied source.

For every $n\in\mathbb N$, does there exist a prime $p$ such that $n^2<p<(n+1)^2$? [S2]

Short English statement

Is there always a prime strictly between the squares of two consecutive positive integers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3100068 (AMR-030-0068). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Jonathan Sorenson and Jonathan Webster, An algorithm and computation to verify Legendre's conjecture up to $3.33\cdot10^{13}$ (2024), Abstract and Section1, Legendre’s conjecture, page1. https://arxiv.org/abs/2401.13753.