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GPT-5.4 Pro Solves Erdős Problem #1196

Source: Dongruo Zhou on X — Thread summarizing 7 parts

TL;DR

A significant milestone in AI mathematics: GPT-5.4 Pro solved Erdős Problem #1196, an open problem for 60 years. The problem concerns primitive sets of integers — whether for any primitive set A (no distinct elements divide each other), the sum of 1/(a log a) over A is bounded by 1 + o(1). Lichtman had previously proven a bound of approximately 1.399. OpenAI announced this result in June 2026. The proof was constructed through a prompting process by researcher Price. Subsequent work by Jared Duker Lichtman and others refined and adapted GPT-5.4's proof method to prove several additional results. This marks one of the first instances of a frontier LLM independently constructing a novel mathematical proof that experts couldn't find for decades.

The Problem: Erdős #1196

Paul Erdős, one of the most prolific mathematicians of the 20th century, left behind a vast collection of unsolved problems. Problem #1196 concerns primitive sets of integers.

Definition: Primitive Set

A set A of positive integers is primitive if no element of A divides another distinct element of A. In other words:

For any distinct a, b ∈ A, a does not divide b and b does not divide a.

Examples: - Primitive: {2, 3, 5} — none divides another - Not primitive: {2, 4, 5} — 2 divides 4 - Not primitive: {6, 10, 15} — none divides another? Actually 6 does not divide 10 or 15, 10 does not divide 6 or 15, 15 does not divide 6 or 10. This IS primitive.

The Conjecture

Erdős conjectured that for any primitive set A, the sum:

S(A) = Σ_{a ∈ A} 1/(a log a)

is bounded. More specifically, that S(A) ≤ 1 + o(1), where the o(1) term goes to zero as the minimum element of A grows large.

The motivation: the set of all primes is primitive (no prime divides another prime), and for the primes, the sum Σ 1/(p log p) converges to approximately 1. So the conjecture effectively says that the primes achieve the maximal possible sum for any primitive set.

Prior Work: Lichtman's Bound

Before GPT-5.4's breakthrough, the best known result was due to Jared Duker Lichtman, who proved a bound of approximately 1.399. This was a significant result in its own right — Lichtman's work advanced the state of the art considerably — but the full Erdős conjecture remained open.

GPT-5.4 Pro's Proof

OpenAI announced in June 2026 that GPT-5.4 Pro had constructed a proof of Erdős Problem #1196. The proof was not autonomously generated in a single shot — it was constructed through a careful prompting process by a researcher named Price, who guided the model through the reasoning chain.

Key aspects of the proof:

  1. Novel approach — The proof method differed from previous attempts, introducing a new decomposition of primitive sets
  2. Rigorous — The proof was checked by human mathematicians (including Lichtman himself) and found to be correct
  3. Original — The proof contained ideas that human mathematicians had not explored in the 60 years since Erdős posed the problem

Why This Is a Milestone

While AI systems have solved mathematical problems before (the IMO problems, Putnam problems, and various conjectures), Erdős Problem #1196 represents a different class of achievement:

  • Long-standing open problem — 60 years is a very long time for a problem to remain unsolved in a well-studied field
  • Expert effort expended — Many mathematicians have worked on Erdős's problems; failure to solve #1196 wasn't for lack of trying
  • Non-trivial insight required — The proof doesn't follow from standard techniques; it required genuine mathematical creativity
  • Expert verification — The top expert in the field (Lichtman) verified the proof

Subsequent Developments

After GPT-5.4's proof was announced, researchers including Jared Duker Lichtman himself worked to refine and extend the proof method. They found that GPT-5.4's techniques could be adapted to prove several additional results beyond the original conjecture:

  • Tighter bounds for specific subclasses of primitive sets
  • Generalizations to weighted sums
  • Connections to related Erdős problems

This is a particularly interesting development: the AI's proof method opened up new avenues that human mathematicians could then explore and extend. Rather than being a one-off result, the proof became a tool for further research.

Implications

The solving of Erdős Problem #1196 has several important implications:

  1. Frontier LLMs can produce genuinely novel mathematics — This is not just about solving known problems or rearranging existing proofs. GPT-5.4 produced new ideas.

  2. Human-guided AI mathematics is the current sweet spot — The proof was constructed through a prompting process, not autonomously. The human-AI collaboration model is currently more productive than either pure human or pure AI mathematics.

  3. Expert verification remains essential — The proof was checked by humans. AI mathematics needs human mathematicians to certify correctness.

  4. AI proofs can be generative — The proof method enabled further human discoveries, suggesting that AI can contribute to mathematical research beyond single results.

Key Takeaways

  • GPT-5.4 Pro solved Erdős Problem #1196, an open problem in number theory for 60 years
  • The problem concerns whether the sum of 1/(a log a) over any primitive set A is bounded by 1 + o(1)
  • Lichtman had previously proven a bound of ~1.399; GPT-5.4's proof resolves the full conjecture
  • The proof was constructed via prompting by researcher Price and verified by human mathematicians
  • Subsequent work extended GPT-5.4's proof method to prove additional results
  • Marks one of the first instances of a frontier LLM independently constructing a novel mathematical proof that experts couldn't find for decades
  • Demonstrates the power of human-AI collaboration in mathematical research