GPT-5.4 Pro Solves Erdős Problem #1202 from Ben Green's List
The GPT-5.4 Pro model, in collaboration with mathematician Liam Price, has solved Problem #1202 from Paul Erdős's problem database. This is Problem 44 from Ben Green's "100 Open Problems," by the leading expert in additive combinatorics and coauthor of the Green–Tao theorem. For the first time, AI has breached Green's "green list"—a curated selection of key open problems. The result is negative: Erdős was wrong in his intuitive conjecture.
The problem was posed in 1980: for k primes p_i, we forbid half the residue classes modulo p_i. The question is whether the set {1, 2, ..., n} collapses almost to zero when k is large enough, but with p_i up to n^{1-ε}? Erdős expected a positive answer, noting the difficulty of proving it.
Erdős Problem Statement
Consider k primes p_1, ..., p_k. For each p_i, choose a subset A_i of {0, 1, ..., p_i-1} with |A_i| = floor(p_i / 2). Define S as the intersection of all
S = { m ∈ {1,...,n} | ∀i, m mod p_i ∈ A_i }.
Erdős conjectured: with a suitable choice of A_i and k ~ log log n / log log log n, the set S has |S| = o(n). For p_i < √n, this follows from the large sieve theorem. But for the larger range p_i ≤ n^{1-ε}, a proof eluded him.
Counterexample from GPT-5.4 Pro and Price
The model and mathematician constructed a counterexample: with p_i on the order of √(n log n) and special forbidden residue classes, |S| reaches (1/2 - c)n—linearly many elements. The construction relies on intervals where the surviving set contains a long arithmetic progression.
Key elements of the counterexample:
- Primes p_i ≈ √(n log n).
- Forbidden classes A_i symmetric with respect to intervals.
- The proportion of surviving numbers doesn't drop below a constant, not o(1).
This refutes Erdős's conjecture. The status of the problem on erdosproblems.com has been updated to "resolved in the negative."
Previous AI Successes on Erdős Problems
The Erdős database contains hundreds of open problems. AI has already solved dozens of them, of varying significance:
- Graph theory problems (e.g., on Ramsey numbers).
- Set combinatorics problems.
- Additive bases and progressions.
The project wiki tracks contributions from AlphaProof, Lean, and other systems. Solving a problem from Green's list is a milestone: his problems were selected as the most promising for progress in the field.
Ben Green, whose theorem with Terence Tao describes arithmetic progressions in primes, curated the list to focus on additive combinatorics. The GPT-5.4 Pro breakthrough demonstrates the potential of large language models in formal mathematics.
What Matters
- Negative result: |S| ≥ (1/2 - c)n with p_i ~ √(n log n), counterexample using intervals and APs.
- First from the "green list": Green selected 100 key problems; AI had previously only tackled the general Erdős database.
- Methodology: Human-AI collaboration—Price verified the model's construction.
- Broader context: AI successes on 30+ Erdős problems are accelerating progress in combinatorics.
- Status: Problem #1202 closed as unresolvable in the positive sense.
The solution highlights a shift: AI not only verifies hypotheses but constructs counterexamples in unresolved areas. For developers, it's a signal to integrate LLMs into proof assistants like Lean or Isabelle.
— Editorial Team
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