Dice Question Streamline Icon: https://streamlinehq.com

Legendre’s conjecture on primes between consecutive squares

Prove that for every integer n ≥ 1 there exists a prime number p with n^2 < p < (n+1)^2.

Information Square Streamline Icon: https://streamlinehq.com

Background

The paper connects the construction of prime-representing constants for exponent c = 2 to the availability of primes in short intervals [n, n + n{1/2}]. This classical problem is closely related to Legendre’s conjecture, which posits a prime between consecutive squares.

The author notes explicitly that Legendre’s conjecture remains unresolved and is believed to be extremely difficult, highlighting why the c = 2 case presents particular challenges.

References

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, 30 Apr 2024) in Section 3 (Lemmas and auxiliary results), paragraph discussing Theorem~\ref{Theorem-Matomaki}