OpenAI and Anthropic models advance on long-standing math problems

Artificial intelligence models developed by technology firms are advancing on long-standing mathematical problems that have challenged researchers for centuries. According to reporting from Popular Mechanics, AI research company OpenAI announced that its next model, Astra, resolved or made progress on a series of long-standing mathematical problems, including Connes’s rigidity conjecture, quantum parallel repetition, and Ehrhart’s volume conjecture. Anthropic also announced that an unreleased version of its model Claude made strides on the Riemann hypothesis through the Riemann zeta function.

Despite these computational strides, the Riemann hypothesis remains unsolved, and Anthropic noted that it does not expect the techniques used by Claude to lead to a complete proof. Instead, the work serves as an indicator of the speed of progress in artificial intelligence capabilities. Mathematicians have expressed a mixture of excitement and skepticism regarding these computational developments, according to interviews published by The Verge. This apprehension has prompted calls for formal ethical frameworks to protect research integrity. A team of researchers proposed the Leiden Declaration on Artificial Intelligence and Mathematics, outlining suggested guidelines and a call to action for the wider research community.

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Understanding the Unsolved Math Frontier

The challenges tackled by artificial intelligence sit alongside other famous mathematical puzzles, such as the Collatz conjecture, Goldbach’s conjecture, the twin prime conjecture, and the behavior of the Euler-Mascheroni constant. Many of these foundational questions remain resistant to complete proof despite decades or centuries of human effort and modern computing power. The Riemann hypothesis itself is among the seven Millennium Prize Problems, and it has implications deep into various branches of math. German mathematician Bernhard Riemann described the Riemann hypothesis and the zeta function in 1859 while studying prime numbers and their distribution.

• Riemann Hypothesis: Concerns the zeros of the Riemann zeta function, with significant implications for number theory.
• Collatz Conjecture: Examines dynamical systems through a simple iterative function applied to natural numbers.
• Goldbach’s Conjecture: States that every even number greater than two is the sum of two primes.
• Twin Prime Conjecture: Asks whether there are infinitely many pairs of primes with a difference of two.

The study of dynamical systems could become more robust than anyone today could imagine, but researchers will need to solve the Collatz conjecture for the subject to flourish. Tao's work represents a near-solution to the Collatz conjecture in some subtle ways, but he most likely cannot adapt his methods to yield a complete solution, as he subsequently explained. Meanwhile, Goldbach’s conjecture precipitated from letters in 1742 between German mathematician Christian Goldbach and legendary Swiss mathematician Leonhard Euler. Euler regarded the conjecture as a completely certain theorem although he could not prove it. When looking at larger numbers, they have more ways of being written as sums of primes rather than fewer, but a proof for all numbers eludes mathematicians to this day.

Regarding the twin prime conjecture, researchers study natural numbers and their properties, frequently involving prime numbers. Mathematicians have managed to tackle closer and closer versions of the twin prime conjecture over the years. In 2013, Yitang Zhang at the University of New Hampshire proved that there are infinitely many primes with a difference of 70,000,000. For the last several years, mathematicians have been improving that number down into the hundreds and eventually down to 6, given some subtle technical assumptions. Recently, a preprint paper from the African Institute for Mathematical Science in Ghana claimed to prove the conjecture, though it has yet to be peer reviewed.

Additional challenges include the Birch and Swinnerton-Dyer conjecture, which involves elliptic curves taking the form of functions that cast insight into algebra and number theory. Sphere packing problems range from pure math to practical applications, including the kissing number problem, where mathematicians have slowly whittled possibilities for dimensions beyond three into fairly narrow ranges. In knot theory, researchers apply formal math ideas to knots, and while several computer algorithms have been written in the last 20 years to identify if a tangled mess is truly knotted, the unknotting problem remains computational and is known to be in NP. Furthermore, in the late 19th century, Georg Cantor proved that infinity comes in different sizes, leading to the study of large cardinals. Beyond these, foundational constants like pi and e remain mysterious when combined, such as whether pi plus e is algebraic or transcendental, while the Euler-Mascheroni constant cannot be proven rational despite being calculated to half a trillion digits.

What to Watch Next

• Future peer review of preprint papers claiming progress on historical conjectures, such as the recent work from the African Institute for Mathematical Science in Ghana.
• Adoption and formalization of ethical guardrails such as the Leiden Declaration on Artificial Intelligence and Mathematics across academic institutions.
• Further disclosures from artificial intelligence developers regarding model capabilities in advanced number theory and dynamical systems.