AI Solves Erdős Math Problem Once Thought Beyond Machines
In a breakthrough that’s sending ripples through both the AI and mathematics communities, a leading large language model has reportedly solved a long-standing problem linked to legendary mathematician Paul Erdős. The result—verified by formal tools and human experts—suggests AI is now capable of more than just pattern recognition: it can reason, synthesize, and even innovate in high-level mathematics. For those wondering whether AI can truly “do math,” this development offers a compelling answer.
From Curiosity to Discovery: How It All Began
The story starts with Neel Somani, a software engineer, former quantitative researcher, and startup founder known for probing the limits of AI systems. Over the weekend, Somani decided to test OpenAI’s latest model on an open problem inspired by Erdős—a figure whose hundreds of unsolved conjectures have become a benchmark for mathematical creativity.
He pasted the problem into ChatGPT and walked away. Fifteen minutes later, he returned to find not just an answer, but a complete, structured proof. Using a verification tool called Harmonic, Somani formalized the model’s output—and everything checked out. “I was curious to establish a baseline for when LLMs are effectively able to solve open math problems compared to where they struggle,” Somani explained. What he found exceeded expectations.
Inside the AI’s Mathematical Reasoning
What makes this result so striking isn’t just that the AI arrived at a correct solution—it’s how it got there. The model didn’t rely on brute-force computation or memorized answers. Instead, it engaged in a sophisticated chain of thought, invoking advanced concepts like Legendre’s formula, Bertrand’s postulate, and even the obscure “Star of David theorem.”
At one point, the AI referenced a 2013 MathOverflow post by Harvard mathematician Noam Elkies, who had tackled a related version of the problem. But rather than simply replicating Elkies’ approach, the model crafted a distinct, more comprehensive proof tailored to Erdős’s original formulation. This ability to adapt, reference, and extend existing knowledge signals a leap in AI’s capacity for genuine mathematical insight.
Why the Erdős Problem Matters
Paul Erdős was famous not only for his prolific output but for posing deceptively simple problems that resisted solution for decades. His conjectures often sit at the intersection of number theory, combinatorics, and logic—areas where intuition and creativity are as crucial as technical skill. Because of their elegance and difficulty, these problems have become a kind of “Olympics” for testing new mathematical methods—including AI.
Solving even a variant of an Erdős problem isn’t just a technical win; it demonstrates that AI can navigate abstract, unstructured intellectual terrain. Unlike chess or Go—where rules are fixed and outcomes clear—open math problems have no predefined path to a solution. That the AI could chart its own course marks a qualitative shift in what machines can do.
Verification and the Human-AI Collaboration
Critically, the proof wasn’t accepted on faith. Somani used Harmonic, a formal verification system, to translate the AI’s natural-language argument into rigorous mathematical logic. The fact that it passed formal scrutiny adds significant weight to the claim.
This moment also highlights a new paradigm: human-AI collaboration in theoretical research. Rather than replacing mathematicians, AI is emerging as a powerful co-pilot—one that can brainstorm, cross-reference, and draft proofs at superhuman speed. Experts say such tools could accelerate discovery, especially in fields drowning in literature but starved for fresh insights.
Skeptics Take Note: This Isn’t Just Pattern Matching
For years, critics have argued that large language models merely remix training data without true understanding. While that critique held for earlier generations, this Erdős breakthrough suggests something deeper is happening. The model didn’t regurgitate a known answer; it synthesized multiple strands of mathematical knowledge to construct a novel argument.
Moreover, the solution required strategic choices—deciding which lemmas to invoke, when to apply asymptotic bounds, and how to structure the logical flow. These are hallmarks of expert reasoning, not statistical mimicry. As AI systems gain longer reasoning windows (like the 15-minute “thinking time” Somani allowed), they’re beginning to mirror the iterative, reflective process real mathematicians use.
What This Means for the Future of AI and Math
If AI can reliably tackle open problems in pure mathematics, the implications extend far beyond academia. Secure cryptography, algorithm design, and even quantum computing all rest on deep number-theoretic foundations. Faster progress in these areas could unlock new technologies—or expose hidden vulnerabilities.
More broadly, this milestone challenges our assumptions about machine intelligence. If AI can contribute to fields once considered the exclusive domain of human genius, it forces us to rethink the boundaries between computation and cognition. The next frontier may not be about scaling models bigger, but about refining how they reason, verify, and collaborate with humans.
A New Chapter in Machine Intelligence
This isn’t the first time AI has dabbled in math—systems like Lean and Coq have assisted in formal proofs for years. But those require human-guided scripting. What’s different here is autonomy: an off-the-shelf language model, given only a problem statement, produced a publishable-grade proof with minimal prompting.
As models grow more capable of sustained, multi-step reasoning, we may see AI not just solving puzzles, but posing new ones—identifying patterns too subtle for human eyes, or suggesting conjectures that reshape entire fields. The Erdős problem may be just the beginning.
For now, the math world is watching closely. And for anyone who doubted that machines could ever “think” like mathematicians, the answer may already be written—in code, in logic, and in the elegant language of proof.
