Polynomial-time computability of factoring
The integer factorization problem: Can the prime factorization of a positive integer be computed in polynomial time? We state the problem by asking if Nat.primeFactorsList is polynomial-time computable (assuming typical encodings of ℕ and List ℕ into bitstrings). Reference: Wikipedia
Current state (tracked automatically)
- Formal Conjectures marks the Lean statement
isPolyTime_primeFactorsListas open.
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Formal Conjectures: status of `isPolyTime_primeFactorsList` | research open | Formal Conjectures |
| Formal Conjectures: research statements in the file | 1 open, 0 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. 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-poly-time-functions,
title = {Polynomial-time computability of factoring},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/poly-time-functions}},
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/poly-time-functions)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
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Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
The integer factorization problem: Can the prime factorization of a positive integer be computed in polynomial time?
We state the problem by asking if Nat.primeFactorsList is polynomial-time computable (assuming typical encodings of ℕ and List ℕ into bitstrings).
Reference: Wikipedia
This file formalizes the open problem of whether natural number / integer factorization can be computed in polynomial time on a deterministic classical Turing machine.
More precisely, it formalizes the following statement: Can the prime factorization of a natural number be computed in polynomial time (on a deterministic classical Turing machine)?
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.Wikipedia.IntegerFactorization.
theorem isPolyTime_primeFactorsList : IsPolyTime Nat.primeFactorsListWhat 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
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.