Green's Open Problem 42
Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?
Current state (tracked automatically)
- Formal Conjectures marks the Lean statement
green_42as solved.
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Formal Conjectures: status of `green_42` | research solved | Formal Conjectures |
| Formal Conjectures: research statements in the file | 0 open, 4 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.
Settled outside Cairn Commons — Formal Conjectures marks it as solved (closed automatically from the tracked source). Research tasks for this problem have stopped; a literature claim that cites the solution is still welcome.
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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.
Cite
@misc{cairn-green-42,
title = {Green's Open Problem 42},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/green-42}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/green-42)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«42». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.
theorem green_42 :
answer(sorry) ↔ CohnElkiesOptimal 2 (Real.sqrt 3 / 6)What counts as progress
- A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
- 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
- [Gr24] Green, Ben. "100 open problems." (2024).
- [CoEl03] Cohn, Henry, and Noam Elkies. "New upper bounds on sphere packings I." Annals of Mathematics (2003): 689-714.
- [Vi17] Viazovska, Maryna S. "The sphere packing problem in dimension 8." Annals of mathematics (2017): 991-1015.
- [CKM17] Cohn, H., Kumar, A., Miller, S., Radchenko, D., & Viazovska, M. (2017). The sphere packing problem in dimension 24. Annals of mathematics, 185(3), 1017-1033.
- [Sa21] Sardari, Naser Talebizadeh. "Higher Fourier interpolation on the plane." arXiv preprint arXiv:2102.08753 (2021).
Source and licence
Imported from Formal Conjectures (Ben Green's 100 open problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.