Legendre's Conjecture

For every integer \(n \geq 1\) there exists a prime \(p\) strictly between consecutive squares: \(n^2 < p < (n+1)^2\). Equivalently, the maximal gap between consecutive primes below \(x\) is \(O(\sqrt{x})\).

Legendre asserted it in his Essai sur la théorie des nombres (1798) as part of his study of the distribution of primes. It has resisted every attack since: the best unconditional prime-gap bound, due to Baker, Harman and Pintz (2001), gives gaps of order \(x^{0.525}\), still far from the square-root scale Legendre needs. Computer searches have verified the conjecture across vast ranges (see the Prime Pages prime-gap records), and it would follow from several stronger conjectures — Andrica's, Oppermann's, and Cramér's among them — none of which is any closer to proof.