- OpenAI’s reasoning model disproved the 80-year-old Erdős planar unit distance conjecture with a novel approach.
- Google DeepMind’s Gemini Deep Think achieved gold-medal level at the 2025 International Mathematical Olympiad.
- New evaluation frameworks like Epoch AI’s FrontierMath are emerging as standard benchmarks lose their ability to differentiate leading models.
An OpenAI reasoning model just disproved a conjecture that mathematicians had left open since 1946, and the most striking part is not the result itself, but how the model got there. Rather than gravitating toward the square-grid arrangements that human intuition had long favoured, the model found a structurally different family of point configurations, drawing on algebraic number theory in a way no one had anticipated. External mathematicians have since validated the proof. Tim Gowers, a Fields Medalist, called it a milestone in AI mathematics and one of the first clear cases of an AI independently solving a famous open problem.
AI Cracks Decades-Old Math Problem
Paul Erdős posed the planar unit distance problem in 1946: how many pairs of points in a set can be exactly one unit apart? The conjecture concerns the upper bound on that count. For nearly 80 years, the best human attempts clustered around square-grid-like configurations. OpenAI’s model, without that ingrained geometric intuition, produced an infinite family of point arrangements that improved on the square-grid bound. The approach was novel enough that it surprised the mathematicians who reviewed it.
What makes this more than a curiosity is what it says about how these models engage with open-ended problems. This was not benchmark performance or pattern-matching on known solutions. The model generated a genuinely new mathematical object. Whether that constitutes “understanding” in any deep sense is a separate debate, but the output was original and correct.
Beyond Benchmarks: IMO Gold and Novel Approaches
At the 2025 International Mathematical Olympiad, Google DeepMind‘s Gemini Deep Think solved five of six problems and scored 35 out of 42 points, reaching gold-medal level. That built on the 2024 performance of DeepMind’s AlphaGeometry and AlphaProof systems, which reached silver-medal standard at the same competition.
Gemini Deep Think’s approach differs from simpler chain-of-thought reasoning. Its enhanced mode runs parallel thinking threads, exploring multiple solution paths simultaneously before committing to an answer. The result is less a single line of reasoning than a kind of structured search.
On the MATH-500 benchmark it scored 96.2%, though independent verification of specific benchmark results is ongoing. Evolving Evaluation: From GSM8K to FormalMATH
Evolving Evaluation: From GSM8K to FormalMATH
Standard benchmarks are running out of headroom. GSM8K, which tests multi-step arithmetic, and MATH, which covers competition-style problems, no longer reliably separate leading models as scores push past 90%. New evaluation frameworks are filling the gap, and the direction they are moving is telling.
Apple researchers introduced GSM-Symbolic, which takes standard problems and varies the numerical values, names and surface details while keeping the underlying logic fixed. Models that appeared strong on the originals often struggled with the variants, suggesting the performance was partly driven by pattern recognition rather than reasoning from first principles. Epoch AI’s FrontierMath benchmark goes further, presenting hundreds of unpublished research-level problems; current models solve only a small proportion. FormalMATH is the most demanding, requiring solutions in proof assistants like Lean 4 where the reasoning must be machine-checkable, not just plausible in natural language.
Each of these reflects the same underlying concern: that fluency and mathematical reasoning are not the same thing, and the older benchmarks could not tell them apart.
Architectural Shifts and Persistent Challenges
Most of the recent progress traces back to a few key architectural decisions. Test-time compute, which OpenAI first disclosed publicly with o1-preview in September 2024, allows models to spend additional compute iterating on their own outputs before returning an answer. More compute at inference time consistently improves performance on hard reasoning tasks, which is a different lever than simply scaling the training run.
DeepMind’s AlphaGeometry takes a hybrid approach, pairing a large language model’s generative capacity with a symbolic engine’s deductive precision. The combination solved 25 of 30 IMO geometry problems, outperforming previous automated methods. It is a design philosophy worth noting: raw language model capability and formal verification machinery are not competing approaches, they are complementary ones.
The limitations are real, though. Large language models are probabilistic systems optimised for fluency, not correctness. A small error in an early reasoning step can propagate silently through a long proof. The same problem can yield different answers on different runs. Models do not natively maintain consistent variable definitions or check whether intermediate claims contradict earlier ones. Fine-tuning helps: training Meta’s LLaMA-2 on the MetaMathQA dataset produced MetaMath-70B, which scored 82.3% on GSM8K, above GPT-3.5-Turbo’s result on the same test. But fine-tuning narrows the problem domain; it does not fix the underlying fragility.
Future Trajectories: Collaboration and Discovery
Gowers’s “milestone” framing matters because it comes from someone with standing to make that call. The Erdős result is not AI beating a human at a structured competition, it is a model producing a proof that working mathematicians had not found. That is a different category of result, even if human verification remains essential to confirm it.
Google, through initiatives involving DeepMind and Google.org, is exploring how AI tools can be integrated into active mathematical research. The stated goal is to combine the intuition of domain experts with AI’s capacity to search large solution spaces, opening research directions that neither could reach alone.
For researchers watching the space, the productive question is not whether AI can do mathematics, but which parts of the mathematical workflow it changes first. The evidence so far points to search and conjecture-generation, the parts that benefit from exploring many candidate structures quickly, rather than the verification and exposition steps that still require human judgement. For more coverage of AI research and breakthroughs, visit our AI Research section.
