legendres_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Legendre's conjecture (1798): There is always a prime number between n² and (n+1)² for every positive integer n. One of the Landau problems. Best known result is that the prime gap following n² is O(n^{0.525}) (Baker–Harman–Pintz 2001), but whether it is always < 2n+1 is unproved.

Why this node was retired

The posted statement is

import Mathlib

theorem legendres_conjecture :
    ∀ n : ℕ, ∃ p : ℕ, Nat.Prime p ∧ n ^ 2 < p ∧ p < (n + 1) ^ 2 := by
  sorry

The natural input n may be0 although the source specifies positive n.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Require n≥1.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.