How AI solved a 60-year-old math problem

When Liam Price sat down on a random Monday to feed a 60-year-old math conjecture into ChatGPT, he wasn't trying to rewrite the history of number theory. He was just curious. Yet, that single prompt to GPT-5.4 Pro resulted in a breakthrough that had eluded professional mathematicians for decades. This isn't just a story about a lucky guess; it’s a masterclass in how AI can bypass the cognitive biases that trap human experts.

The problem in question involves "primitive sets"—collections of whole numbers where no element divides another. Paul Erdős, the legendary mathematician, conjectured that the "Erdős sum" of these sets approaches a specific limit as the numbers grow. For sixty years, the brightest minds in the field approached this with a standard sequence of moves, essentially hitting a wall because they were all looking at the problem through the same narrow lens.

Here’s where most people get tripped up: they assume AI is just a fancy calculator. In reality, the AI’s strength here wasn't raw computation; it was lateral thinking. By applying a formula well-known in other branches of mathematics but never before linked to primitive sets, the model effectively "vibe-maths" its way around a mental block that had stalled human progress.

If you’re wondering how an amateur can outpace experts, the answer lies in the lack of baggage. Human mathematicians are trained to follow established paths. When those paths lead to a dead end, we often assume the problem is harder than it actually is. The AI, unburdened by the "standard sequence of moves," simply looked for a different connection.

That said, there’s a catch. The raw output from the model was, by all accounts, a mess. It required human experts like Terence Tao and Jared Lichtman to sift through the noise, identify the core insight, and formalize the proof. This is the future of AI-assisted mathematical research: a symbiotic relationship where the machine provides the creative leap, and the human provides the rigor.

Why does this matter for your own work? It highlights a critical failure mode in expert-led fields: the tendency to over-complicate problems based on historical precedent. If you find yourself stuck on a complex challenge, try reframing it for an LLM. You might be surprised when it suggests a path you’ve been conditioned to ignore.

This new method for solving primitive sets isn't just a one-off win. It validates the intuition that many of these long-standing conjectures are clustered together, sharing a deeper, hidden anatomy. We are moving into an era where the barrier to entry for high-level discovery is shifting. You don't necessarily need a PhD to contribute to the field; you just need the right tools and the willingness to ask the machine the right questions.

The jury is still out on the long-term significance of this specific proof, but the precedent is set. We’ve seen that an amateur armed with ChatGPT can indeed solve a 60-year-old math problem by simply refusing to follow the crowd. Try this approach on your next complex project and share what you find in the comments.