In today’s column, I examine a recently heralded victory by OpenAI in using AI to pursue a longstanding unresolved math problem. Headlines about this accomplishment were proclaiming that AI has become some sort of math genius.
The truth is a bit more down-to-earth. Though there is much in the circumstance that merits applause, the mainstay of the effort was not to provide a positive mathematical proof per se, but instead to discover a counterexample that undercuts a prevailing conjecture regarding the solution to the thorny problem.
I will explain why this is a beast of a different condition. Noteworthy mathematicians were at first a tad disappointed because they thought the AI had derived a heretofore unreached mathematical proof. This would have put AI over-the-top. The AI was actually utilized to laboriously and deeply search for a means of disproving the existing conjecture of the solution. It’s quite a different direction when seeking to disprove an assertion than when generating a proof on a go-ahead basis.
In any case, I doubt many discerned that this was a handy lesson on the myriads of ways to use AI. Rather than always having AI plow forward into a morass, make sure to consider whether you ought to employ AI to tackle a vexing problem from the perspective of discovering make-or-break counterexamples. The advantages can be significant.
Let’s talk about it.
This analysis of AI breakthroughs is part of my ongoing Forbes column coverage on the latest in AI, including identifying and explaining various impactful AI complexities (see the link here).
The Disproving Of A Conjecture
In a recent posting by OpenAI entitled “An OpenAI Model Has Disproved A Central Conjecture In Discrete Geometry” on May 20, 2026, these salient points were made (excerpts):
- “For nearly 80 years, mathematicians have studied a deceptively simple question: if you place n points in the plane, how many pairs of points can be exactly distance 1 apart?”
- “This is the planar unit distance problem, first posed by Paul Erdős in 1946. It is one of the best-known questions in combinatorial geometry, easy to state and remarkably difficult to resolve.”
- “Since Erdős’s original work, the prevailing belief has been that the ‘square grid’ constructions were essentially optimal for maximizing the number of unit-distance pairs. An internal OpenAI model has disproved this longstanding conjecture, providing an infinite family of examples that yield a polynomial improvement.”
- “The proof has been checked by a group of external mathematicians.”
- “The proof came from a new general-purpose reasoning model, rather than from a system trained specifically for mathematics, scaffolded to search through proof strategies, or targeted at the unit distance problem in particular.”
Those points briefly depict the math problem that was being pursued.
General-Purpose AI Is The Hero
A subtle but crucial aspect in that last point is that the AI was apparently not devised for this specific mathematical exercise. In other words, an AI maker could shape an entire AI system that is entirely focused on solving math problems. This is a common tactic.
The AI that was used in this instance was seemingly a general-purpose AI. We don’t know all the details of the inner workings of the AI, but let’s assume the AI is indeed a general-purpose construction. This bodes well for what can be done with general-purpose AI. Perhaps the cost and effort of devising specialized AI will not be needed for some challenging math problems. A somewhat everyday AI could be used. It’s an encouraging sign if it holds up for other math problems on hand.
My analogy would be that a car mechanic might customarily require a highly specialized and expensive tool to fix cars. They then discover that an ordinary hammer will do the same. Voila, start using hammers. This even opens the path of fixing cars to everyday citizens, since having access to hammers is relatively easy and inexpensive.
Proving Something Versus Disproving Something
As noted, the crux of the AI usage was that there already existed a human-devised conjecture about the math problem, and the AI was directed to find a counterexample to disprove the conjecture. Let’s mull this over by considering the act of proving something versus disproving something.
Suppose a friend of yours tells you that they have never stated on their social media that unicorns exist. They are absolutely sure they haven’t ever declared on social media that unicorns are in existence. They stake their reputation on this.
You ask your friend to prove that they have never mentioned the existence of unicorns. What would the friend need to do? Well, they would start collecting all their social media postings. One by one, they would show you that there wasn’t any mention of unicorns in any of those postings.
The Directions You Can Go
Are you convinced by your friend’s listing of their social media postings?
Probably not, because there might be more social media postings that the friend hadn’t perchance rounded up. You might contend that there are possibly more of their social media postings to be found and inspected. On and on this might go. It could be never-ending.
Here’s an angle you might go. If you want to undercut your friend, all you need to do is find just one single instance of them mentioning the existence of unicorns on their social media. All that is required is one such instance. By finding a counterexample concerning their conjecture, you have disproved their claim.
Which direction then is best?
Your friend is proceeding with the pursuit of examining all their social media postings. It’s a high burden. There might always be an elusive one that they didn’t perchance find. On the other hand, if you opt to simply find one instance that disproves their conjecture, you win. And you might win without nearly as exhaustive and costly an effort undertaken.
Mathematicians Speak Up
Various mathematicians looked at the OpenAI work on the math problem. The AI had produced a rather voluminous chain-of-thought output on what took place. You might be curious about what the math experts had to say.
In a posting that OpenAI linked to entitled “Remarks On The Disproof Of The Unit Distance Conjecture” by Noga Alon, Thomas Bloom, W.T. Gowers, Daniel Litt, Will Sawin, Abul Shankar, Jacon Tsimerman, Victor Wang, and Melanie Wood, May 20, 2026, these salient points were indicated (excerpts):
- “The proof that we give in these remarks is a human-digested, somewhat simplified, and somewhat generalized version of the AI proof.”
- “The idea of trying to use number fields to construct counterexamples is not altogether new, but there are subtleties to making it work (discussed in some of the reflections in this note), especially considering that perhaps most experts believe the conjecture n**(1+o) to be true.”
- “As a result, it is likely that most of the human efforts spent on this problem have been on trying to prove the upper bound, rather than spending serious time on trying to disprove it.”
- “If the result of this paper were a proof of the unit distance problem, that would be truly incredible. While I was still very surprised to hear of this result, this was dampened slightly when I learnt it was a construction of a counterexample, and still further when I learnt the nature of the construction, being (with the benefit of hindsight) a natural, albeit highly non-trivial generalization of the original lattice-based construction of Erdos.”
There are two key takeaways there.
One is that headlines and word-of-mouth were at times implying or suggesting that the AI had found a final proof to the problem. The proper way to portray this is that AI was able to disprove the conjecture about the solution to the problem. Those are radically different considerations.
The second takeaway is that humans were apparently focused on trying to prove the conjecture, rather than disproving it. That’s an intriguing notion. When humans might all be focusing their attention on solving a problem one way, it might be handy to employ AI to try to address the problem from a different angle.
Business Value Is Notable
Let’s ruminate on that precept of aiming to use AI to disprove or find a counterexample, rather than solely being used to prove something outright.
A financial company has instituted a fraud detection capability. They are confident that this will curtail all chances of fraud being committed. The executives are ecstatic.
Inside the firm, some are not quite so sure that this fraud detection capability is unassailable. They ask for proof that it will indeed catch all possible fraud attempts. An extensive effort by company engineers and analysts is launched to try to prove that it handles everything under the sun.
Meanwhile, an employee uses AI to see if an instance can be found that disproves the claim that the fraud detection covers all possible bases. This is often referred to as red-teaming AI. Rather than proving that a system works perfectly, the AI is directed to search for plausible counterexamples.
Sure enough, the AI finds an edge condition that manages to sneak under the fraud detection threshold. This disproves the conjecture. Note that this result doesn’t necessarily mean that the fraud detection should be junked. There could be modest patches that would shore up handling the found counterexample.
Embrace Both Sides
Counterexamples are great and can be exciting.
But be wary of anyone who says that the best way to always proceed is to find a counterexample or aim to disprove something. Finding a single counterexample can be powerful and destroy an entire conjecture, but the lack of finding a counterexample doesn’t prove that the conjecture must be right. Sometimes finding a counterexample is extraordinarily difficult.
The bottom line is that there is not a mutually exclusive arrangement going on. You ought to pursue the proof side and pursue the counterexample side. Coming at a problem from only a one-sided perspective is usually insufficient. As the old saying goes, there is more than one way to skin a fish.
With any challenging problem, we could rely on Albert Einstein’s famous remark: “If I had an hour to solve a problem, I’d spend 55 minutes thinking about the problem and 5 minutes thinking about the solutions.” For those who are trying to solve vexing problems, you might want to take his advice to heart. In addition, invoke AI to help you, though make sure the AI is pointed in the right direction, or at least in a productive direction.
