A Claimed Computer-Checked Proof of Conway's 50-Year-Old Conjecture

Dan Abramov says he believes he has a Lean proof of a conjecture John Conway posed 50 years ago. The claim appeared Sept. 18, 2026, on overreacted.io, Abramov's personal blog, in a post titled 'How I Vibed a Proof of Conway's Conjecture' overreacted.io. That post is the original source for the claim.

The post defines the target as Conway's refinement conjecture for omnific integers. The claim is that if ab = cd then there exist integers e, f, g, h with a = ef, b = gh, c = eg, and d = fh overreacted.io. The definition turns on that factorization shape.

On validation, Abramov makes two explicit statements. He states the proof has not been independently verified by mathematicians. He also states it passed the mechanical checks from Palomar registry entry PALOMAR-2026-09-03-000002 version 1 overreacted.io.

The broader context here is the gap between machine checking and mathematical acceptance. A kernel verdict confirms that the formal text elaborates and checks under a pinned toolchain and registry entry. It does not by itself settle whether the definitions capture the intended informal mathematics. People who work with proof assistants daily treat that separation as routine. It is also where review should concentrate. Definitions deserve close reading. Proof scripts deserve replay under independent setups.

Looking at what this means for practitioners, the workflow suggested by the title is the part to study. Interactive proving with language models in the loop shortens the time between a conjectured lemma, a small helper claim, and a kernel response. That loop is productive. It also shifts the dominant failure mode. Instead of getting stuck on a tactic block, a unit of proof automation, the risk becomes specification drift, where the version that checks is subtly different from what readers assume was proved. Pinned registry checks improve reproducibility. They leave semantic review untouched. Experienced formalizers inspect implicit coercions, typeclass instances, and library definitions before trusting a green checkmark.

In my view, the responsible reading is narrow and still encouraging. A pinned, mechanically checking formalization awaiting human review gives others something concrete to inspect, refute, or tighten. Independent elaboration under the same registry entry can confirm reproducibility quickly. Careful reading of definitions and the refinement witnesses will take longer. That work is normal. I have watched new tooling move from demo to daily use, and the lasting value often lies less in any single claimed result than in the reusable patterns it leaves behind. If this formalization survives scrutiny, practitioners gain both a theorem and a template. If it does not, the failure trace still teaches something useful about where automated formalization needs stronger guardrails.