Representing a significant milestone in AI-assisted mathematical research, a team at Axiom Math has automatically verified the proof of a theorem relating to prime numbers—colloquially referred to as the “246 theorem”—for the first time using the company’s AI system AxiomProver.
In formal verification, mathematicians task a computer with checking a machine-readable version of a proof. The process is not a 100 percent guarantee that the proof is correct, as a recent demonstration showed, exposing how a bug in the method could be exploited to accept a false, AI-generated proof. Still, the computational method is as close to a rubber stamp as you can get.
This particular verification formalizes an important advance in number theory. Beyond this particular proof, it demonstrates how automated AI verification could be used in the future to ensure the correctness of AI-generated computer code that will soon underlie software across the globe.
This is not AxiomProver’s first rodeo. Axiom Math has used its autonomous, multi-agent system that turns mathematical statements into machine-checkable proofs to crack several unsolved mathematical problems and verified many more proofs this year. But proof formalization of the 246 theorem is by far the most significant, as Ken Ono, Axiom Math’s founding mathematician, explains: “This theorem currently represents the threshold of human knowledge about prime numbers.”
Earlier this year, Axiom Math competitor Math, Inc. used its Gauss agent to formalize Maryna Viazovska’s 2022 Fields Medal-winning proof of the sphere-packing problem in 8 and 24 dimensions. Sidharth Hariharan, a Ph.D. student at Carnegie Mellon University who led human efforts on the blueprint to formalize Viazovska’s proof, says that formalizing the 246 theorem is a more comprehensive and useful achievement.
Now an intern at Axiom Math, Hariharan has been heavily involved in the company’s formalization of the 246 theorem proof. He says that one of the main differences here is that rather than it being a one-shot approach relating to a single problem, Axiom Math has expressly aimed to make components of the formalization reusable for other formalization tasks and mathematical research. The team has wielded AxiomProver to build a library of results about gaps in primes. The 246 theorem is the flagship result within that library.
The first few primes are close together: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, .... And there are several instances where they are separated by a difference of two: 3:5, 5:7, 11:13, 17:19, ...
These pairs of primes are called twin primes. Twin primes become rarer the further you get from zero, but they do still seem to pop up occasionally. The twin prime conjecture, first precisely formulated in the 19th century by French mathematician Alphonse de Polignac, posits that they will keep popping up regardless of how far along the number line you look. In other words, there are infinitely many twin primes.
Though easy to state, the venerable twin prime conjecture remains unproven. First progress toward solving it only occurred in 2013 when Yitang Zhang, now a professor at Sun Yat-sen University, in Guangzhou, China, proved that there are infinitely many pairs of primes that are separated by 70 million. A few months later, using a different technique, University of Oxford professor James Maynard dramatically reduced this gap from 70 million to just 600; a feat which substantially contributed to Maynard being awarded the 2022 Fields Medal—widely regarded as the Nobel Prize for mathematics.
As part of a group of mathematicians known as the Polymath8b collaboration, Maynard and fellow Fields Medalist Terence Tao, professor at the University of California, Los Angeles, brought the gap down to just 246; the closest mathematicians have gotten to the target gap of two. It is this 246 theorem—which states that there are infinitely many primes that differ by 246—that AxiomProver has verified to be correct.
The techniques formalized in this work are important in number theory, the branch of mathematics that underpins all present-day cybersecurity and cryptography. They could therefore prove to be useful in verifying specific ways in which we keep our digital data safe in the future.
But Axiom Math’s Ono is more excited by the bigger picture. He sees formalizing mathematical proofs as a stepping stone to verifying AI-generated code, which is starting to be used across society in systems that run our infrastructure, manage our finances, and protect our data. This is despite safety concerns surrounding hallucinations, bugs, and other unintended vulnerabilities.
If properties of code—such as whether an algorithm terminates or if a program’s output is correct for any input—can be translated into precise mathematical statements, technologies derived from AxiomProver would be ideally suited to formally stating and proving them. In this way, mathematically verifying the correctness of AI-generated code would make this code safe to use. “The world is about to run on computer code that nobody has read,” Ono concludes. “AI is here and we can no longer look away—proof formalization is a testbed for solving what I think is the most important challenge we will face from AI.”