OpenAI had generated plenty of buzz by coming up with a breakthrough for an 80-year-old Erdős problem, but Anthropic now says that it has also come up with a solution with its Mythos model.
An Anthropic researcher posted over the weekend that the company’s Claude Mythos model — the powerful, not-yet-publicly-released frontier model best known for its cybersecurity capabilities — had independently solved the planar unit distance problem, the same 1946 Erdős conjecture that OpenAI cracked just days earlier. Anthropic’s head of alignment science Sholto Douglas added his own comment, calling it evidence of “serious overhang in discoveries.”
What The Problem Is
The unit distance problem is easy to state: if you scatter n points on a flat surface, what is the most pairs of points you can have that are exactly distance 1 apart? Paul Erdős posed the question in 1946 and conjectured that the answer was, essentially, not much more than the number of points themselves. For nearly eight decades, the best anyone could do — using a cleverly rescaled grid of integer points — barely budged the needle. OpenAI’s model was the first to prove Erdős wrong, showing that you can do substantially better. Anthropic says Mythos found its own proof of the same thing.
A Simpler Route To The Same Destination
What’s notable about the Mythos solution, according to the Anthropic researcher’s post, is its relative elegance. The OpenAI proof was a conceptual leap that imported ideas from a highly abstract branch of math — algebraic number theory — in a way that surprised the mathematics community. Mythos apparently arrived at a solution along the same general path but with a more streamlined argument.
The core idea: take a number field (a structured extension of ordinary arithmetic) with certain properties, and build an infinite tower of related fields on top of it. Each field in the tower has a ring of integers — a generalization of whole numbers — and within the field formed by adjoining i (the imaginary unit) to each of these, there is a group of special elements called norm-one units. These units, when projected down to the real plane via an embedding, give directions in which unit distances can be arranged. The key is that the number of such directions grows faster — much faster — than what the classical grid construction allows.
Crucially, the Mythos version stays entirely within what the researcher called “the literal dumbest thing”: add points of size at most half the radius to units of size at most half the radius, and count the unit-distance pairs you get. The geometry-of-numbers machinery — which translates the algebraic structure of the field into counts of lattice points — kicks in immediately and handles the rest, with no additional analytic machinery required.
The attached preprint formalizes the argument. It shows that for infinitely many values of n, the maximum number of unit-distance pairs grows as n raised to a power strictly greater than 1 — disproving Erdős’s conjecture with an exponent that improves on the grid construction by a factor involving iterated logarithms.
Arriving After A Flurry Of Activity
The timing is notable. OpenAI’s breakthrough on the unit distance problem drew validation from Fields Medalist Tim Gowers, who said he would have recommended the proof for publication in the Annals of Mathematics without hesitation. Within days, Princeton professor Will Sawin had already improved on the result by hand, pushing the explicit exponent from roughly 6×10⁻³⁸ to 0.014. Now Anthropic says its own model had independently found a path to the same conclusion.
The convergence matters. In science, independent replication of a result is a strong signal that something real has been found. Two frontier AI systems arriving at structurally similar proofs for the same open problem — without coordination — is a meaningful data point about the reliability and depth of what these models can do.
It also fits the broader pattern. Mythos has already demonstrated exceptional performance on mathematical reasoning benchmarks, and Anthropic’s researcher noted the company had set up a dedicated testing environment for Erdős problems following earlier AI math breakthroughs. The unit distance problem was dropped into that environment with internet access blocked, to rule out retrieval of known results.
What This Signals
Sholto Douglas’s comment about “serious overhang in discoveries” is worth taking seriously. The implication is that AI models capable of this kind of mathematical work have been sitting in front of these problems for a while — and now that the ice is broken on AI solving long-standing open questions, the results may come faster than anticipated.
Mathematics is particularly well-suited to validate this: the answers are verifiable, the problems are well-defined, and there is no ambiguity about what counts as a solution. If two frontier models, tested independently, both find valid proofs of the same 80-year-old conjecture in the same week, it says something meaningful about the depth of capability these systems now possess — and how much of that capability is only beginning to be systematically applied to hard problems.