OpenAI just used its new large language model GPT-5.6 Sol to solve a math problem that humans have struggled with for more than a half-century. And all it took was telling the artificial intelligence not to give up.
The company timed the proof of the “cycle double cover conjecture” to the model’s full public release last Friday. The achievement shows how OpenAI and other AI firms are heavily investing in pure mathematics as a way to benchmark the technology’s ascent towards reasoning—and the strange directions those efforts often take.
“It is a well-known conjecture, which received a considerable amount of attention over the years—and surprisingly, the proof is short,” says Noga Alon, a mathematician at Princeton University. Alon calls the breakthrough “yet another impressive example demonstrating that AI tools will change—and are already changing—mathematical research significantly.”
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The conjecture is about graphs. A graph is just a bunch of dots (called “vertices”) connected by lines (called “edges”). Mathematical proofs about graphs have been applied to all kinds of real-world networks, including the Internet. In the 1970s several different mathematicians guessed that almost any graph you can imagine will have a “cycle double cover.”
A cycle is just a loop within a graph that ends where it started; a “cycle double cover” is a special set of loops that cover the entire graph exactly twice—each edge is part of exactly two loops. Over the years, a number of prominent mathematicians tried to prove this guess of their forebears. And prove it they did—for a number of specific cases, anyway, but never in general.
Last Friday’s AI-generated proof seems to have settled the question. It shows that any guess-applicable graph can be doubly covered with no more than eight well-chosen loops (for technical reasons, graphs with big sections connected by a single edge—like twin cities with a single road between them—are excluded). The proof, like other notable AI achievements in math, required surprisingly little in the way of new ideas. It just followed and combined methods humans had tried before and managed to squeeze a bit more out of them. As AI companies continue to scour the mathematical literature for open problems that their models can solve, they seem to keep digging out “easy” quandaries disguised as hard ones—conjectures for which a proof was always within reach but somehow never grasped by humans.
Once a problem gets a reputation for being “hard,” experts and students might spend less time on it, says Andrew Sutherland, a mathematician at the Massachusetts Institute of Technology. This can make perceived difficulty a kind of self-fulfilling prophecy. “My guess is we will keep seeing examples of this—supposedly ‘hard’ problems having ‘easy’ solutions found by LLMs,” he says.
OpenAI also released the prompt that led to the successful proof. It reveals some of the quirky tricks that engineers have discovered for coaxing chatbots to grapple with math. “Most of that prompt is scaffolding aimed at getting the LLM to actually put in the effort needed to solve the problem,” Sutherland says. The prompters directed their model to delegate its tasks to as many as 64 agents working in parallel. Combining efforts of many AIs in communication with one another is a known way to mitigate false proofs and hallucinated references. They also told it not to simply dismiss the problem as unsolved and hence unlikely to yield a solution. Perhaps most importantly, they gave it some firm encouragement: “Spend at least 8 hours on this before even thinking of returning or giving up.”
This aligns with many mathematicians’ anecdotal experiences with LLMs, in which the AI will try to wriggle out of finding novel proofs by noting humans’ failed attempts. The right approach seems to be something like a schoolteacher’s—a mix of affirming praise and stern directions.
