Legendre’s conjecture (primes between consecutive squares)

Prove that for every integer n≥1 there exists at least one prime number in the interval (n^2,(n+1)^2), thereby establishing Legendre’s conjecture.

Background

In explaining the construction of prime‑representing constants when c=2, the paper notes the strong connection to finding primes in intervals of length n{1/2}. This is intimately related to Legendre’s conjecture on primes between consecutive squares.

The authors explicitly remark that this conjecture remains unresolved and is considered extremely difficult, underscoring the challenge of pushing constructions analogous to Mills’ approach to the c=2 case.

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 Discussion following Theorem-Matomäki, Section 3 (Lemmas and auxiliary results)