GPT-5.4 Pro Closes a 1968 Erdős Problem With a Method Mathematicians Overlooked for 90 Years
On April 13, 2026, a 23-year-old with no formal mathematics training solved Erdős problem #1196, a conjecture posed in 1968 that had resisted professional mathematicians for nearly six decades. Liam Price wrote a single prompt to GPT-5.4 Pro. The model returned the exact answer in 80 minutes.
The problem concerns primitive sets: collections of whole numbers where no element divides another. Paul Erdős proved in 1935 that the sum of 1/(a·log a) over any primitive set is always finite. Problem #1196 asked how small that sum can become when restricted to large numbers. Stanford mathematician Jared Lichtman — who had spent seven years on this exact problem and held the previous best bound of approximately 1.399 — described the AI’s result as the first proof at the level of Erdős’s Book.
The Book, in Erdős’s parlance, was the imaginary volume God keeps containing the most elegant proof of every theorem. GPT-5.4 Pro produced the exact asymptotics: 1+O(1/log x).
The Method Is the Story
What matters is not that the problem was solved. It is how.
Since 1935, every mathematician who touched #1196 translated it from number theory into probability theory. That conversion felt so natural, so structurally obvious, that no one looked for an alternative. GPT-5.4 Pro built the proof through the von Mangoldt function — an object from analytic number theory that encodes the fundamental theorem of arithmetic. Lichtman called it a new opening line in chess that human convention had simply never considered.
The comparison that has spread through mathematical circles: AlphaGo’s Move 37 in 2016. The move looked like a mistake. It rewrote the theory of Go.
Within 24 hours, Fields Medallist Terence Tao had taken Price’s result and extended it into the seeds of a new mathematical theory.
What Has Changed
This is not the first time AI has been credited with Erdős solutions. In October 2025, OpenAI claimed GPT-5 had solved ten problems; the math community pushed back hard, calling it a misrepresentation of literature-retrieval as discovery. Thomas Bloom, who runs erdosproblems.com, called that episode a dramatic misrepresentation.
Problem #1196 is different in character. The previous human-best bound (Lichtman, 2023) was publicly available and well-known. The AI did not find an obscure reference. It found a different path. Lichtman, who had dedicated years to the problem, validated the result immediately and used the Book comparison without qualification.
Scientific American covered the result on April 24. Terence Tao’s public commentary on Mathstodon has been careful but unambiguous: the method appears genuinely novel, and the extension he is building from it may open new lines of research.
What It Does Not Mean
GPT-5.4 Pro solved one problem from a catalogue of over 1,100. The problems range from tractable to potentially unsolvable with current mathematics. The AI did not autonomously identify which problem to target, design the research programme, or verify the formal proof — that last step required Harmonic’s Aristotle system to translate the natural-language argument into Lean, the computer-verifiable proof language. Price wrote the prompt.
The right framing: a highly capable reasoning model, given a precise formulation by someone who understood what to ask, found a path through a problem that human intuition had been systematically missing. That is significant. It is not yet a general-purpose mathematical collaborator.
The Broader Pattern
Erdős problems have become a practical benchmark for AI mathematical capability because they are stated simply, their difficulty is well-understood, and the community has an existing infrastructure — Bloom’s erdosproblems.com — to track progress and call out overclaims. More than a dozen problems have been marked solved since December 2025, with AI involvement acknowledged in most cases.
The qualitative shift with #1196 is the methodological novelty. Earlier AI results mostly applied known techniques to underexplored formulations. This one introduced a tool from a different branch of mathematics that the relevant experts had not considered. That is a harder claim to dismiss.
Key Numbers
| Metric | Detail |
|---|---|
| Problem | Erdős #1196, posed 1968 |
| Human best (2023) | ~1.399 upper bound (Lichtman) |
| AI result | 1+O(1/log x) exact asymptotics |
| Time to solve | 80 minutes, single attempt |
| Method | von Mangoldt function (analytic number theory) |
| Validation | Jared Lichtman (Stanford), Terence Tao (UCLA) |
| Model | GPT-5.4 Pro via ChatGPT Pro subscription |