Source labels openChecked July 26, 2026

Oppermann's Conjecture

Oppermann's Conjecture*: For every integer $x \ge 2$ , the following hold: - There exists a prime between $x(x-1)$ and $x^2$ . - There exists a prime between $x^2$ and $x(x+1)$ .

Number theory AMS 11

Mathematical statement

Oppermann's Conjecture*: For every integer $x \ge 2$ , the following hold:

There exists a prime between $x(x-1)$ and $x^2$ .
There exists a prime between $x^2$ and $x(x+1)$ .

Statement source: Wikipedia statement material

Statement terms: CC-BY-SA-4.0

Attributed source material. Reuse must follow the linked attribution and share-alike terms.

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

oppermann_conjecture

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem oppermann_conjecture (x : ℕ) (hx : 2 ≤ x) :    (∃ p ∈ Ioo (x * (x - 1)) (x^2), p.Prime) ∧    (∃ p ∈ Ioo (x^2) (x * (x + 1)), p.Prime) := 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

Pinned Formal Conjectures sourceformalization Wikipediabackground Luan Alberto Ferreira, *Real exponential sums over primes and prime gaps*background