A long-standing mathematical conundrum has been solved by ChatGPT in a few hours, with only a few simple prompts.
The Dinitz-Garg-Goemans conjecture is a 30-year-old question in graph theory, but a posted on X by , co-founder at AI start-up Autokernel, has shown that it is false.
Rybin entered just four prompts into ChatGPT 5.6 Pro: an initial one instructing the AI to 鈥渄o a breakthrough and find a structured counterexample鈥�, followed by three others simply urging it to continue searching. All four prompts added up to fewer than 60 words, and the AI took a total of 5.5 hours to crack the problem.
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Graph theory is the study of networks made up of nodes, or vertices. The Dinitz-Garg-Goemans conjecture can be thought of as a logistical challenge: imagine shipments from a warehouse to multiple locations can be split into much smaller deliveries that can be sent on different routes. The conjecture states that this scenario can be converted into another where shipments cannot be split, and that the total cost of shipping will not increase.
Rybin didn鈥檛 respond to a request for comment, but : 鈥淚 know counterexamples to old conjectures are becoming a meme at this point. But I really cared about this problem and spent many weeks thinking about it.鈥�
聽at the University of York, UK, says there鈥檚 a running joke in mathematics that every conjecture in graph theory is false, just as the Dinitz-Garg-Goemans conjecture has now been proved to be.
鈥淚n fields like number theory or algebra, patterns that hold for small rank [simple situations] often hold for a long time,鈥� says Bownman-Scargill. 鈥淲hereas in graph theory, structural behaviour can shift dramatically once you add just one or two vertices鈥� which is how these conjectures continue to be made. You can see how people miss these things.鈥�
AI has made rapid advances in mathematics in recent months. In May, an OpenAI model聽cracked a decades-old conjecture by Paul Erd艖s, causing a stir in mathematical circles. Earlier this week, an AI found a counterexample to the Jacobian conjecture, which had stood for nearly a century. Today, other AI users claim to have solved a and a . A website has even sprung up to and list them by the model that was used.
at Queen Mary University of London says current AI models seem particularly well-suited to problems like the Dinitz-Garg-Goemans conjecture, but there are limits to what is currently possible 鈥� and the problems solved by AI so far are of limited complexity.
鈥淎I is really good, and at least at some mathematical tasks, already superhuman,鈥� says Saha. 鈥淭here is a fair bit of low-hanging fruit out there. Some conjectures can now be proved or disproved by AI with very little human input; the main challenge is simply pointing the system in the right direction. On the other hand, I don鈥檛 think AI is yet at a place where it can build the theory needed to prove some of the deepest open conjectures people care about.鈥�
But there are signs that AI is here to stay and will become a vital tool for mathematicians. at the University of Illinois Urbana-Champaign says AI鈥檚 growing role in mathematics will empower researchers to discard dead ends and instead push in promising directions.
鈥淐ounterexamples to old conjectures are never聽quite聽as impressive as finding a sequence of interlocking arguments that constitute a proof,鈥� says Yong. 鈥淗owever, I鈥檇 expect that AI will soon prove many conjectures by a combination of their inherent superhuman energy in knowing the literature and trying many things at a prompt.聽Those conjectures that survive AI scrutiny will be the genuine goals for human innovation.鈥�
