Open problemChecked July 24, 2026

Legendre's Conjecture

A prime-gap question asking for a prime between every pair of consecutive squares.

Number theory Prime numbers Prime gaps

Mathematical statement

For every positive integer $n$ , there is a prime $p$ satisfying $n^2<p<(n+1)^2$ .

Statement source: Therefore editorial statement

Statement terms: Source-specific

Editorial wording by Therefore. No statement reuse license is granted.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

LegendreConjecture.legendre_conjecture

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem legendre_conjecture :    answer(sorry) ↔      ∀ n ≥ 1, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2),        Nat.Prime p := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Landau's problemsbackground