Integer factorization proven subexponential, with core work done by Claude
burny_tech · x · 2026-10-30
A new result establishes a deterministic, worst-case subexponential bound for integer factorization: a deterministic Turing machine can find the smallest prime factor of any N in N^o(1) steps (within Ck·2^(n/k) for n-bit inputs). The algorithm reduces factoring to counting divisors with prescribed leading bits, leveraging the counting engine from the ETH refutations.
- Comes with a Lean certificate: the statement to check is only 13 lines using just Mathlib; the full proof is 35,000 lines, written less than 24 hours after the ETH refutations appeared.
- The provenance statement says the ideas and Lean proof were generated by Claude Opus 5.5 (Anthropic); the authors only specified and audited the formal statement — the proof itself is checked by Lean.
- Prior best deterministic bound was N^(1/5+o(1)); Anthropic's original ETH refutation proof runs 152,000 lines, and this paper re-proves only the counting-engine part it needs.
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