Alexander Rimov
How a mathematician from Nizhny Tagil and ChatGPT found a counterexample that had eluded researchers for nearly thirty years.
Until recently, the idea that a language model could contribute to a genuine mathematical discovery belonged more to science fiction than to serious research.
That assumption was shattered this week by an unlikely collaboration between a mathematician from Russia’s industrial heartland and an artificial intelligence.
The story centers on Dmitry Rybin, an independent mathematician from the city of Nizhny Tagil, who, together with ChatGPT, accomplished something that had resisted the efforts of specialists for almost three decades: he disproved one of the most persistent conjectures in modern algebraic combinatorics.
Yet the most remarkable part of the story is not the result itself.
It is how the result came about.
ChatGPT did not replace mathematical reasoning.
It expanded it.
Instead of serving as a faster calculator or a more knowledgeable reference book, the model became an intellectual sparring partner—one capable of proposing ideas that experienced researchers would almost instinctively dismiss as absurd.
A Puzzle That Lasted Three Decades
The challenge revolved around the Dinitz–Garg–Goemans conjecture, proposed in the mid-1990s. They concerned a family of graphs arising in group theory and asserted that, under a specific collection of structural constraints, every graph in the family must contain a Hamiltonian cycle—a closed path that visits every vertex exactly once before returning to its starting point.
To non-specialists, the problem appeared esoteric. Within algebraic combinatorics, however, it occupied an important place, with implications reaching into graph algorithms and cryptographic theory.
Over the next thirty years, the conjectures acquired an almost mythical status.
More than a hundred papers produced results consistent with them. Millions of individual cases were tested on high-performance computers. Every experiment pointed in the same direction.
The consensus gradually became almost unquestioned.
Surely, researchers believed, only the final elegant proof remained to be found.
As it turned out, the real obstacle was never the missing proof.
The conjectures themselves were wrong.
A Conversation That Changed the Rules
Dmitry Rybin was not part of the academic establishment.
A graduate of a regional teachers’ college, he spent his days teaching mathematics at a vocational school while pursuing independent research on graph-theoretic counterexamples.
When OpenAI released a new generation of ChatGPT with dramatically improved reasoning capabilities, Rybin saw an opportunity that few others had imagined.
Most researchers were treating large language models as sophisticated assistants—useful for summarizing papers, checking calculations, or generating code.
Rybin had a different idea.
What if ChatGPT were used not as a tutor, but as what he would later describe as “a generator of heresy”?
So he asked the wrong question on purpose.
Instead of requesting a proof of the Dinitz–Garg–Goemans conjecture, he supplied ChatGPT with their formal assumptions and posed an entirely different challenge.
“Imagine you are an alien intelligence with no intuition shaped by human mathematics. Construct the ugliest, most asymmetric, most counterintuitive graph that satisfies every assumption of these conjectures while behaving in the strangest possible way.”
The answer looked like nonsense.
The output was little more than a tangled collection of vertices and edges, apparently devoid of structure or mathematical elegance.
Most mathematicians would have dismissed it within seconds.
Rybin didn’t.
He translated the construction into adjacency matrices and began examining it piece by piece.
The longer he looked, the stranger it became.
Hidden beneath what appeared to be mathematical chaos was a graph with 257 vertices exhibiting an extraordinarily rare symmetry—one that disguised itself so effectively that exhaustive searches over smaller instances had never exposed it.
The graph satisfied every assumption of the Dinitz–Garg–Goemans conjecture.
Except for one devastating detail.
It had no Hamiltonian cycle.
Putting the “Ugly Graph” to the Test
Rybin’s first reaction was disbelief.
Not excitement.
Not triumph.
Disbelief.
If the construction was correct, it would overturn nearly thirty years of accepted mathematical intuition. That was precisely why he assumed it had to be wrong.
For several days he searched for the mistake he was certain must be hiding somewhere. Every edge was checked. Every matrix was reconstructed. Every structural property was verified independently.
Then he rebuilt the graph from scratch in Wolfram Mathematica.
The result refused to change.
There was no hidden error.
The ugly graph that had emerged from less than a minute of conversation with ChatGPT turned out to be exactly what generations of mathematicians had failed to find: a genuine counterexample—and, remarkably, the smallest one theoretically possible.
Rybin prepared a manuscript, submitted it to a specialist journal, and uploaded the preprint to arXiv.
The reaction was immediate.
Within days, researchers around the world began reconstructing the graph independently.
Among the first was a group at the Massachusetts Institute of Technology, whose members had spent years searching for a proof of the Dinitz–Garg–Goemans conjecture.
By Monday morning, the verdict was unanimous.
The construction was valid.
The counterexample held.
The Dinitz–Garg–Goemans conjecture were dead.
More Than a Tool
Rybin’s work reaches far beyond a single counterexample.
It points toward something much larger: a new way of doing science.
For years, artificial intelligence has been viewed primarily as an extraordinarily efficient computational tool. Systems such as AlphaFold transformed biology by predicting protein structures with unprecedented speed, but their role remained fundamentally computational.
ChatGPT played a different game.
It did not outperform human expertise.
It challenged human intuition.
Rather than searching for elegant proofs, it explored bizarre possibilities—structures so awkward, asymmetric, and counterintuitive that most mathematicians would never have considered them worth investigating.
In that sense, the model became something closer to an advocatus diaboli—a devil’s advocate relentlessly proposing ideas that human aesthetic instincts would reject before logic had the chance to evaluate them.
As one mathematics professor at the Higher School of Economics observed in a widely shared social media post:
“For thirty years we were prisoners of elegance. We kept looking for harmony where none existed. The AI has no sense of mathematical beauty. It simply explores logical possibilities. And precisely because it lacks our aesthetic instincts, it discovered a monster we had dismissed as impossible.”
Whether that diagnosis ultimately proves correct is almost beside the point.
The episode has already forced mathematicians to confront an uncomfortable possibility.
Perhaps creativity in mathematics is not always about seeing beauty.
Sometimes it begins by being willing to imagine ugliness.
What Comes Next?
Today, Rybin is working on a follow-up paper—not about graph theory itself, but about the process that led to the discovery.
Its subject is what he calls prompt engineering for abstract algebra: a systematic method for using large language models not to automate mathematical thinking, but to provoke it.
The distinction matters.
ChatGPT did not replace the mathematician.
It became an extension of his imagination.
Instead of spending years exploring countless dead ends through brute-force computation, Rybin used the model to venture into regions of the mathematical landscape that human intuition rarely visits. The machine did not supply certainty. It supplied possibility.
That may prove to be the deeper lesson of this story.
For generations, mathematics has celebrated elegance. Beautiful proofs, elegant constructions, and natural symmetries have guided researchers toward some of the discipline’s greatest discoveries. Yet beauty can also become a form of intellectual gravity, quietly pulling generations of mathematicians toward the same assumptions—and away from possibilities that feel too strange to deserve attention.
Artificial intelligence has no such instincts.
It does not know which ideas are supposed to be beautiful.
Nor does it know which ones are supposed to be impossible.
That ignorance may turn out to be one of its greatest strengths.
The implications reach far beyond a single conjecture.
How should journals evaluate papers in which the crucial insight emerges from a conversation with an AI? What does authorship mean when the decisive idea comes from a machine incapable of understanding its own reasoning? Should creativity itself still be regarded as an exclusively human faculty?
Those questions are no longer philosophical thought experiments.
They have become practical problems for science.
The answers will no doubt evolve over time. But one principle is unlikely to change.
Science has never cared where an idea comes from.
It cares only whether the idea survives every attempt to destroy it.
Rybin’s counterexample did.
The Dinitz–Garg–Goemans conjecture did not.
The future has arrived.
It has an IP address.
Yuri Chekalin
Nikita Parshunin
