Legendre's conjecture: a prime between every two consecutive squares

Known gap bounds are too weak. Baker, Harman and Pintz (2001) proved a prime exists in $[x - x^{21/40}, x]$ for all large $x$, so gaps of size $p^{0.525}$; Legendre needs gaps below about , i.e. exponent . Cramér showed gaps are , still a logarithm short. Cramér's heuristic gap size would give Legendre for large . a table of maximal prime gaps confirms the conjecture up to . a 2023 arXiv preprint by Luan Alberto Ferreira (revised May 2026, submitted to , not published) claims that every interval with contains asymptotically primes, which would give Legendre for all sufficiently large (the Formal Conjectures file labels the large- statement as solved, citing Ferreira). It has not been confirmed by independent review; since the Riemann Hypothesis gives such asymptotics only for , the claim goes beyond RH and needs especially careful checking. Even if correct, it would not settle small beyond the numerical range.

None directly; primes in short intervals matter for the analysis of algorithms that search for primes, but nothing deployed depends on the exponent. The methods (zero-density estimates and exponential sums) are reused elsewhere in analytic number theory.

A Lean proof of legendre_conjecture as stated. A proof of the eventual version alone (for all sufficiently large $n$) is a genuine partial result and would be recorded as PARTIAL, with the finite range closed by computation checked in Lean.