AI Tackles Unsolved Math Problems

Author: Denis Avetisyan

A new automated pipeline demonstrates that artificial intelligence can now reliably solve sophisticated, research-level mathematical problems.

State-of-the-art large language models, integrated with citation-based verification, achieve success on challenging problems from the First Proof Problem Set.

Despite recent advances in artificial intelligence, automating the solution of open research-level mathematical problems remains a significant challenge. This work, ‘Can a Lightweight Automated AI Pipeline Solve Research-Level Mathematical Problems?’, demonstrates that state-of-the-art large language models, integrated into a streamlined, citation-verified pipeline, can reliably generate solutions to sophisticated mathematical problems, including those previously unpublished. We evaluated this approach on novel datasets comprising both competition-level problems and original research questions, successfully verifying solutions for a substantial subset. Could this represent a viable pathway toward widespread AI assistance in mathematical discovery and formalization?

The Fragility of Formal Systems

Despite their impressive ability to generate human-like text, current large language models consistently falter when confronted with complex mathematical problems. The difficulty isn’t simply a lack of factual knowledge; rather, these models struggle to construct and verify extended chains of reasoning-the very foundation of mathematical proofs. While proficient at identifying patterns within datasets, they often fail when required to apply those patterns in novel situations or to rigorously justify each step in a solution. This limitation stems from the models’ training methodology, which primarily focuses on statistical relationships between words rather than the logical coherence necessary for mathematical deduction. Consequently, even seemingly straightforward problems requiring multiple interconnected inferences can expose the fragility of their reasoning capabilities, hindering their potential as reliable tools for mathematical exploration and discovery. The inability to reliably generate [latex]step-by-step[/latex] proofs represents a significant barrier to advancing AI’s role in mathematical research.

Conventional language model training relies heavily on exposure to vast quantities of text, yet this approach falls short when applied to mathematical proofs. These proofs aren’t simply about recognizing patterns in text; they demand a deep understanding of abstract concepts, logical relationships, and the implicit rules governing mathematical systems. A model trained solely on raw text learns to correlate words, but fails to grasp the why behind each step in a derivation. For instance, understanding [latex] \nabla \cdot (\nabla \times \mathbf{F}) = 0 [/latex] requires more than just seeing the equation; it necessitates grasping the principles of vector calculus and the relationship between divergence and curl. Consequently, models struggle to generalize beyond memorized examples, hindering their ability to independently construct or verify complex mathematical arguments, and limiting their potential as true assistants in mathematical discovery.

The escalating ambition to utilize artificial intelligence in mathematical research compels a fundamental evolution in AI design, moving beyond superficial pattern recognition. Current AI excels at identifying correlations within datasets, but genuine mathematical discovery demands a capacity for deductive reasoning – the ability to rigorously construct and verify proofs. This isn’t simply about processing more data; it requires imbuing AI with the capability to understand the underlying logic of mathematical statements, to explore abstract concepts, and to generate novel insights based on established axioms. Consequently, the future of AI in mathematics hinges not on scaling existing machine learning techniques, but on developing systems capable of symbolic manipulation, formal verification, and ultimately, exhibiting a form of computational creativity – a shift from ‘knowing’ to ‘understanding’ the language of mathematics, enabling it to tackle unsolved problems and potentially uncover new [latex]\mathbb{ theorems [/latex]].

Architecting for Originality: An Automated Pipeline

The automated pipeline is engineered to address mathematical problems requiring originality and proof construction, surpassing the scope of competitive mathematics such as the International Mathematical Olympiad. Current systems often excel at well-defined problems with known solution pathways; this pipeline aims for research-level tasks demanding novel approaches and the ability to formulate and validate conjectures. This includes problems in areas like number theory, topology, and combinatorics where existing datasets are insufficient for supervised learning, necessitating a generative approach to problem-solving and proof generation. The system’s performance is evaluated not only on correctness but also on the originality and non-triviality of its solutions, as determined by expert mathematicians.

The automated pipeline utilizes large language models (LLMs) Gemini 3 Pro and GPT-5.2 Pro to generate and evaluate potential mathematical proofs and solutions. These next-generation LLMs are selected for their demonstrated capacity in complex reasoning tasks and their ability to produce novel outputs based on provided inputs. The integration involves formulating mathematical problems as prompts for the LLMs, processing the generated responses, and employing verification mechanisms to assess the validity of the solutions. This leverages the LLMs’ generative power to explore a broad solution space, exceeding the capabilities of traditional automated theorem proving systems which often rely on pre-defined rules and search algorithms.

Prompt optimization within the automated pipeline utilizes techniques including iterative refinement, chain-of-thought prompting, and the strategic inclusion of relevant contextual information to improve the Large Language Model’s (LLM) ability to tackle complex mathematical problems. This process moves beyond simple query formulation, focusing instead on structuring prompts to elicit step-by-step reasoning and the application of abstract mathematical principles. Specifically, the system employs automated methods to test variations in prompt phrasing, length, and the inclusion of guiding examples, evaluating the LLM’s output against established mathematical correctness criteria. The goal is to maximize the LLM’s performance on problems requiring not just computational skill, but also the identification of appropriate theorems, the construction of logical proofs, and the manipulation of [latex]\mathbb{Z}[/latex]-level abstract concepts.

Tracing the Logic: Citation-Augmented Verification

The citation-augmented verification method mandates that the model explicitly provide bibliographic references for each non-trivial claim made within a mathematical proof. A claim is considered non-trivial if it is not a basic axiom, a previously established theorem within the current proof, or a definition. This requires the model to identify the source – typically a theorem, lemma, or established result – that justifies each step in the derivation. The citation must uniquely identify the referenced material, enabling external verification of the claim’s validity and grounding the proof in existing mathematical literature. Failure to provide a valid citation for a non-trivial claim results in a verification failure.

Requiring citation for each non-trivial claim within a mathematical proof significantly mitigates the risk of model hallucination by anchoring assertions to verifiable sources. This process enforces a dependency on established mathematical literature, effectively constraining the model to knowledge explicitly documented in cited works. By demanding bibliographic support for each step, the system reduces the generation of unsubstantiated statements and provides a traceable lineage for every logical inference, thus increasing the reliability and trustworthiness of the generated proofs. The system operates by cross-referencing claims against the cited literature to confirm the validity of each step; discrepancies are flagged as potential errors or hallucinations.

Validation of the citation-augmented verification method employed Kashiwara’s ‘Categories and Sheaves’ as a benchmark text. During testing, the method successfully identified assertions within proofs that lacked corresponding bibliographic support within the text. Specifically, any claim not directly traceable to a definition, lemma, or theorem explicitly cited from ‘Categories and Sheaves’ was flagged as unsubstantiated. This demonstrated the system’s capacity to not only require citations for non-trivial claims, but also to verify the existence of those citations within a defined corpus, thereby reducing the likelihood of accepting logically unsound or hallucinated proof steps.

Mapping the Mathematical Landscape: Performance and Impact

Rigorous evaluation of the automated reasoning pipeline involved testing against established benchmarks and, crucially, a newly constructed dataset of unsolved problems. The International Mathematical Olympiad Contest Problems (ICCM) provided a foundation for assessing performance on well-defined challenges, but the true measure of the system’s capabilities lay in its ability to address genuinely novel mathematical questions. This ‘First Proof’ set, comprised of research-level problems not previously published, offered an unprecedented opportunity to gauge the pipeline’s capacity for original thought and problem-solving in uncharted territory. Successful navigation of this challenging dataset demonstrates a leap beyond simple pattern recognition, indicating an ability to engage with the creative process inherent in mathematical discovery.

The framework exhibits a remarkable aptitude for tackling challenges across diverse mathematical domains, extending beyond standard problem-solving to include the nuanced task of identifying counterexamples within the Analytic Theory of Polynomials. This capability signifies a departure from merely confirming existing theorems; the system actively probes for conditions where established mathematical principles might fail, a crucial aspect of rigorous mathematical inquiry. Complementing this, the pipeline demonstrates proficiency in solving complex combinatorial optimization problems – scenarios involving a vast number of possible configurations where the goal is to identify the optimal solution. Such problems, frequently encountered in fields like logistics, computer science, and operations research, require navigating immense solution spaces, and the framework’s success highlights its capacity for efficient and accurate computation in these challenging contexts.

The framework demonstrates a unique ability to engage with the intricacies of Category Theory, a highly abstract branch of mathematics concerned with relationships between mathematical structures. This capacity isn’t merely about processing symbols; it indicates a potential for reasoning about mathematical concepts at a fundamentally different level. Category Theory often underpins advanced areas like theoretical computer science and mathematical physics, and successful navigation of its principles suggests the framework can move beyond concrete calculations to manipulate and explore abstract relationships – a key step towards automating higher-level mathematical discovery. The successful handling of these problems points towards a system capable of not just solving mathematics, but of understanding and potentially generating new mathematical insights within abstract formalisms.

Rigorous testing demonstrates the pipeline’s exceptional problem-solving capabilities, achieving a perfect score on both established ICCM problem sets. This signifies a landmark accomplishment in automated mathematical reasoning, as the pipeline successfully navigated and resolved every challenge presented within these datasets. The consistent, flawless performance isn’t merely a quantitative result; it validates the underlying architecture’s capacity to accurately interpret, strategize, and execute solutions across a diverse range of mathematical problems. This complete success rate establishes a new benchmark for automated theorem provers and signals a substantial advancement in the field of artificial intelligence applied to complex mathematical domains.

The framework’s capabilities were further validated through its application to a novel ‘First Proof’ problem set, comprising ten currently unpublished, research-level mathematical challenges. Successful completion of this set signifies not merely the reproduction of known solutions, but the pipeline’s ability to independently derive solutions to problems unseen by existing automated theorem provers. This accomplishment demonstrates a capacity for genuine mathematical reasoning, extending beyond pattern matching and symbolic manipulation to encompass the creative problem-solving inherent in original mathematical discovery. The ten solved problems span diverse areas of mathematics, suggesting a broad applicability and a robust foundation for tackling future, even more complex, challenges.

The Inevitable Convergence: AI and Mathematical Research

As mathematical knowledge accumulates, the challenge isn’t primarily finding new theorems, but ensuring the absolute correctness of existing and newly proposed proofs. This shift in focus demands a move beyond human review and necessitates the development of robust formal methods – systems capable of mechanically verifying mathematical reasoning. These methods translate mathematical statements into a precise, unambiguous language that a computer can analyze, checking each step against established axioms and inference rules. Currently, researchers are exploring various logical frameworks, including proof assistants like Coq and Isabelle, and automated theorem provers, to build these verification systems. Successfully automating proof verification will not only bolster confidence in mathematical results but also unlock the potential for AI to rigorously examine and extend complex mathematical arguments, significantly accelerating the pace of discovery and ensuring the reliability of increasingly intricate theoretical structures – for example, validating proofs in areas like number theory, where [latex]p=NP[/latex] remains unproven.

A significant hurdle in leveraging artificial intelligence for mathematical research lies in the inherent incompleteness of published mathematical arguments. While mathematicians often omit steps considered obvious, these implicit logical chains are inaccessible to AI systems requiring explicit reasoning. Consequently, advancements in natural language processing and machine learning are focused on reconstructing these missing links within mathematical literature. This involves not just identifying stated theorems and definitions, but also inferring the underlying assumptions and reasoning patterns employed by mathematicians. Successful development of these capabilities would allow AI to verify the correctness of proofs, identify potential errors, and ultimately, provide complete reasoning justifications for every step-transforming how mathematical knowledge is processed and validated. Furthermore, the ability to automatically fill in these gaps is essential for AI to not only check existing work, but also to independently explore and generate new mathematical insights, potentially accelerating the rate of discovery in the field.

The potential for artificial intelligence to move beyond simply verifying existing mathematical proofs and toward genuine discovery represents a paradigm shift in how research is conducted. Current efforts focus on enabling AI systems to not just check for errors, but to formulate and test conjectures, explore potential avenues of proof, and ultimately, generate novel mathematical insights. This involves developing algorithms capable of identifying patterns, extrapolating from existing knowledge, and creatively combining different mathematical concepts – skills previously considered exclusive to human mathematicians. By automating aspects of the creative process, these AI systems promise to dramatically accelerate the pace of mathematical research, allowing mathematicians to focus on the most challenging and nuanced problems, and potentially unlocking solutions that might otherwise remain elusive for generations. The integration of AI isn’t intended to replace human intellect, but rather to augment it, creating a collaborative environment where machines and mathematicians work in synergy to expand the boundaries of mathematical knowledge.

The pursuit of automated mathematical reasoning, as detailed within this research, isn’t about achieving flawless calculation-it’s about establishing a dynamic system capable of iterative refinement. This work suggests a pipeline, not a solution. A system that never breaks is, indeed, a dead one. Paul Erdős famously said, “A mathematician knows all there is to know.” While this pipeline doesn’t approach omniscience, it embodies the spirit of relentless exploration. The ability to verify citations, crucial to this approach, acknowledges the inherent imperfections within any formalized system, allowing for a continuous process of correction and growth. It’s a propagation of ideas, not a perfect monument to logic.

What Lies Ahead?

This work doesn’t so much solve research-level mathematical problems as it shifts the locus of difficulty. The pipeline demonstrates a capacity for formalizing and verifying existing knowledge, but it reveals, with stark clarity, the fragility of the foundations upon which that knowledge rests. A system isn’t a machine, it’s a garden – and this pipeline, for all its automation, merely tends the borders of the known, exposing the wilderness beyond. The true challenge isn’t building a proof engine, but cultivating a landscape where interesting questions naturally bloom.

The reliance on citation-based verification, while pragmatic, invites a subtle, yet significant, dependency. The system isn’t reasoning from first principles, but rather tracing paths through a pre-existing web of assertions. Resilience lies not in isolation, but in forgiveness between components – a failure in one link doesn’t necessarily break the chain, but it does highlight the need for richer, more nuanced understandings of mathematical truth. Future work must explore methods for generating genuinely novel insights, not just rearranging established facts.

Perhaps the most pressing question isn’t whether these systems can do mathematics, but whether they can ask the right questions. The automation of deduction is a valuable tool, but it’s a tool nonetheless. The real progress will come when these systems begin to exhibit a form of mathematical intuition – a capacity for recognizing patterns, formulating conjectures, and, crucially, knowing when to abandon a fruitless line of inquiry. A system can polish a theorem, but only a mind can conceive one.

Original article: https://arxiv.org/pdf/2602.13695.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/