Mathematician Paul Erdős died nearly 30 years ago, but left behind a massive legacy. Famously known for his collaborations with other mathematicians and his eccentric practice of showing up unannounced to their doorsteps and staying only as long as they provided him with interesting challenges to ponder, the prolific mathematician left behind 1,135 problems for succeeding mathematicians to advance the field by either proving or disproving them.
In the last few months, three of those problems have been cracked by Artificial Intelligence (AI) systems harnessed with the well established Lean proof assistant. NeuronDaily tells the story of the third, which is impressive because there was little-to-no online material the AI system could have accessed to develop its proof, which means the AI paired with a proof assistant is generating genuinely original work:
Remember those brain-teaser math problems from school that made you want to throw your pencil? Now imagine ones so hard they've stumped mathematicians for decades. Paul Erdős, who published more papers than anyone in math history, left behind hundreds of these puzzles when he died in 1996. This weekend, GPT-5.2 Pro solved one.
Neel Somani prompted the AI to tackle Erdős Problem #397, which asks whether infinitely many solutions exist for a specific equation involving central binomial coefficients. GPT-5.2 generated the proof, the tool Aristotle formalized it in Lean (a verification language), and Fields Medalist Terence Tao accepted it.
Here's Neel Somani's X tweet announcing the accomplishment:
Weekend win: The proof I submitted for Erdos Problem #397 was accepted by Terence Tao.
— Neel Somani (@neelsomani) January 11, 2026
The proof was generated by GPT 5.2 Pro and formalized with Harmonic.
Many open problems are sitting there, waiting for someone to prompt ChatGPT to solve them: pic.twitter.com/hnUBQ7YCBp
NeuronDaily explains why the proof of Erdős Problem #397 is a big deal:
It’s part of a wave of autonomous solves: GPT-5.2 has now cracked Problem #728, #729, and 397.
Verified mathematics: The Aristotle system auto-corrected gaps in proofs and produced Lean-verified code.
Self-contained reasoning: Unlike October 2025's GPT-5 controversy (which just found existing literature), Tao says these are original proofs.
While impressive for the technology, the new proofs don't yet represent the kind of ground-breaking accomplishment that would permanently leave human mathematicians in the dust.
The catch? Tao emphasizes these are “lowest hanging fruit”; problems solvable with standard techniques, not profound breakthroughs. GPT-5.2 scores 77% on competition-level math but only 25% on open-ended research requiring genuine insight.
Three years ago, having any computing system score 77% on competition-level math would have been a remarkable achievement, but scoring 25% on open-ended research was a pipe dream. Considering the amazing progress that's been made with the use of AI in maths in just the past three years, what lies in store for the next three years?
Image credit: Paul Erdős and Terrence Tao in 1985. Wikimedia Commons. Creative Commons Attribution-Share Alike 2.0 Generic Deed. We were surprised that Erdős collaborated with Tao when the latter was just 10 years old! Terence Tao, of course, went on to have an equally remarkable career and has an Erdős Number of 2.