Pre-release GPT-6 Astra disproves Erdős's 'first serious problem' (1931, $500) and proves the rational-exponents conjecture, all Lean-verified, in Epoch's FrontierMath Erdős runs

2026-09-03★★★★★after cutoffscienceOpenAIEpoch AIconfidence: high

In Epoch AI's FrontierMath Erdős runs (announced 3 Sep 2026), a pre-release GPT-6 Astra autonomously resolved five of 68 hand-picked open Erdős problems with Lean-checked proofs. They include a disproof of Erdős problem #1 on distinct subset sums, which Erdős dated to 1931 and called 'perhaps my first serious problem', and a proof of the Erdős–Simonovits rational-exponents conjecture for bipartite Turán numbers (#571). The Erdős–Sós conjecture (#548) is in a separate entry.

Key facts

Benchmark: 68 open Erdős problems chosen by Thomas Bloom from 652 open problems on erdosproblems.com; a resolution counts only if it is proved in Lean; default budget $300 and 72 hours per problem
Official score (one attempt each): pre-release GPT-6 Astra 3% (#74 disproved for $218 in 15 h; #126 proved for $247 in 16 h); GPT-5.6 Sol, GPT-5.5, Claude Fable 5.1 and Claude Fable 5 all 0%
Extra, non-benchmark attempts with larger budgets resolved 5 problems in total (#1, #74, #126, #548, #571), using over $220,000 of compute versus about $20,000 for the benchmark run
#1 (Erdős, dated 1931, $500 prize): if A ⊆ {1..N} has n elements with all subset sums distinct, must N ≫ 2^n? Astra showed that for every ε > 0 there are arbitrarily large n with N ≤ ε·2^n, so the conjecture is false. The previous best construction was N ≤ 0.22002·2^n (Bohman). The proof is ineffective (no explicit n for a given ε). Found in 2 of 4 attempts ($405 / 27 h and $1,384 / 84 h)
#571 (Erdős–Simonovits): for every rational α in [1,2) there is a bipartite graph G with ex(n; G) ≍ n^α. Bloom calls it 'the most difficult of the five solutions'. Found in 1 of 3 attempts ($617, 41 h)
#74 (Erdős–Hajnal–Szemerédi): disproved in the unexpected direction. If every n-vertex subgraph can be made bipartite by deleting at most f(n) edges with f growing slowly enough, the graph has chromatic number ≤ 3
#126 (Erdős–Turán): |S(A)| ≫ n^(1/2) primes divide the pairwise sums of an n-element set, improving the classical log n bound; Astra gave three distinct proofs (exponents 1/8, 1/3 and 1/2)
erdosproblems.com now marks #1 as 'DISPROVED (LEAN)'
The Lean repositories (tadamcz/erdos1, erdos74, erdos126, erdos571), created 3 Sep 2026, say the proofs were found autonomously in the benchmark harness; the report calls the informal write-ups 'placeholders' until human experts prepare proper papers

Science result

Field: mathematics / additive combinatorics / extremal graph theory
Problem: Erdős problems #1 (distinct subset sums), #74, #126 and #571 (rational exponents conjecture) (open since 1931)
Result: #1 disproved (N ≤ ε·2^n possible for arbitrarily large n); #571 proved (every rational α in [1,2) is a Turán exponent of a bipartite graph); #74 answered in the negative direction; #126 proved with |S(A)| ≫ n^(1/2).
AI system: GPT-6 Astra (pre-release)
Human role: Autonomous: the model worked in an agent harness without human steering; humans chose the problems, wrote the Lean statements and wrote informal expositions afterwards
Verification: Formal proofs in Lean 4 checked against audited challenge statements; informal human digestion ongoing; not peer-reviewed
Status: confirmed
Why surprising: The oldest open Erdős problem, which Erdős posed at 18, fell to an autonomous agent run as a side effect of building a benchmark.

What happened

Epoch AI and Thomas Bloom (erdosproblems.com) built FrontierMath Erdős, a benchmark of 68 open Erdős problems formalised in Lean, and ran five frontier models on it. Under the fixed $300 budget only the pre-release GPT-6 Astra solved anything (2 of 68). In further runs with larger budgets, which the authors report separately and do not count as benchmark results, Astra also disproved problem #1, proved the Erdős–Sós conjecture (#548) and proved the rational-exponents conjecture (#571). Every resolution comes with a Lean proof checked against a small challenge statement.

Why it matters

Problem #1 is probably the longest-standing open Erdős problem. The #571 result settles a central question on Turán numbers of bipartite graphs. The paper also gives a denominator that most AI-math announcements lack. The fixed-budget run solved only 2 of 68 problems. Of the 63 unsolved problems, 56 were retried two to five times each (172 attempts) with no success, and the five resolutions took over $220,000 of compute. That makes it a sober counterweight to OpenAI's later claim of "100+ open problems".

Changelog

2026-09-30: created

Related events

2026-09-03GPT-6 Astra proves the Erdős–Sós conjecture (1962) with a short counting argument; mathematicians race to simplify and extend it ★★★★★
2026-09-03OpenAI releases GPT-6 Astra, its first GPT-6 model ★★★★★
2026-09-10GPT-6 Astra's Epoch AI run adds more Lean-checked results: Dittert conjecture proved, Ibragimov–Iosifescu and eternal-domination conjectures disproved ★★★
2026-09-07Pre-release GPT-6 Astra disproves the Köthe conjecture (1930) with a Lean-verified counterexample ★★★★
2026-09-21OpenAI says an internal model resolved 100+ long-standing open problems in 24 days of training; no list released ★★★
2025-12-08Genuine AI-assisted solutions to Erdős problems begin: #124 (Aristotle), #1026 (48-hour human–AI collaboration) ★★★★
2026-09-30Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) ★★★★

Sources (8)

id: 2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1 · updated 2026-09-30 · open in the interactive timeline