Legendre’s Conjecture: Prime Between Consecutive Squares

Determine whether, for every positive integer x, there exists at least one prime number p with x^2 < p < (x+1)^2.

Background

Legendre’s conjecture asserts the existence of a prime in every interval between consecutive squares. It is a classical open problem in the distribution of primes and directly relates to bounding gaps between primes.

In the paper’s context, the author surveys classical conjectures on prime gaps as motivation. Subsequent claimed bounds on prime gaps would imply results about primes in short intervals near squares.

References

For example it has been conjectured by Legendre that given a positive integer $x$, there is at least one prime number between $x2$ and $(x+1)2$, Oppermann(1877) made a slightly stronger conjecture that given a positive integer $x$, there is at least one prime number between $x(x-1)$ and $x2$, and a prime between $x2$ and $x(x+1)$.

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

This problem is strongly connected with Legendre's conjecture which asserts that for every n there exists a prime number between n2 and (n+1)2. It is unsolved and believed to be extremely difficult.

— Mills' constant is irrational  (2404.19461 - Saito, 2024) in Discussion following Theorem-Matomäki, Section 3 (Lemmas and auxiliary results)