Deterministic integer factoring
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Given an integer N in binary, how quickly can one deterministic algorithm find its smallest prime factor? Complexity is measured in the input bit length n, approximately log₂ N, with an explicit finite-alphabet machine model.
Why it matters and what counts as progress
Why it matters: Factoring is a basic problem in computational number theory. A smaller asymptotic bound matters even when enormous constants prevent practical use.
What counts as progress: Prove a better worst-case deterministic bound with exact input/output and machine conventions. Distinguish the smallest factor from complete factorization, randomized methods and quantum algorithms.
How the problem developed
- 2021 · Research result [1]
David Harvey Published a deterministic exponent-one-fifth factoring algorithm, the fixed-exponent baseline cited by the new project.
- 2022 · Research result [1]
David Harvey and Markus Hittmeir Improved logarithmic factors within the exponent-one-fifth deterministic bound.
- 26 Oct 2026 · AI contribution [2]
Claude (Anthropic) Reported a subexponential exact-counting engine. The factoring project later re-proved the part it needed.
- 28 Oct 2026 · AI contribution [1]
Isaac Rudich, Louis-Martin Rousseau and Claude Reported one deterministic smallest-prime-factor machine with time N^{o(1)}, with source-reported Lean checking. The stronger stated conjecture is unproved.
AI contribution
Subexponential Deterministic Factoring
A single deterministic Turing machine outputs the smallest prime factor Nat.minFac N of every natural number N in binary. For each integer k≥1 there is a constant C_k such that its running time on every n-bit input is at most C_k·2^{⌊n/k⌋}; all stack alphabets are finite. Thus the stated deterministic running time is N^{o(1)}.
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