Oppermann's Conjecture

For every integer x ≥ 2 there exists a prime between x(x-1) and x^2.

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	3 open, 1 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-oppermann,
  title        = {Oppermann's Conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oppermann}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

oppermann_conjecture.parts.i. For every integer $x \geq 2$ there exists a prime between $x (x - 1)$ and $x^{2}$ .

oppermann_conjecture.parts.ii. For every integer $x \geq 2$ there exists a prime between $x^{2}$ and $x (x + 1)$ .

oppermann_conjecture. Oppermann's Conjecture: For every integer $x \geq 2$ , the following hold:

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

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Oppermann (3 statements).

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

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

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)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.