/-
Copyright 2026 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
import FormalConjecturesUtilPolynomial-time computability of factoring
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)?
References:
namespace PolyTimeopen ComplexityTheoryThe 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
@[category research open, AMS 68]
theorem isPolyTime_primeFactorsList : answer(sorry) ↔ IsPolyTime Nat.primeFactorsList := ⊢ True ↔ IsPolyTime Nat.primeFactorsList
All goals completed! 🐙end PolyTime