# AI for Mathematics

Three large language models independently reproved all 465 propositions of Euclid's Elements, and their citation choices differ systematically from Heath's human-proof baseline. Mean pairwise Jaccard similarity among three Claude Opus 4.6 runs is 0.68, nearly twice the 0.37 between AI and human proofs, so the difference is stable and reproducible. Its concrete signature is that about 40% of AI citations point to foundational elements such as definitions, common notions and postulates, against only 13% in human proofs.

Three large language models independently reproved all 465 propositions of Euclid's Elements, and their citation choices differ systematically from Heath's human-proof baseline. Mean pairwise Jaccard similarity among three Claude Opus 4.6 runs is 0.68, nearly twice the 0.37 between AI and human proofs, so the difference is stable and reproducible. Its concrete signature is that about 40% of AI citations point to foundational elements such as definitions, common notions and postulates, against only 13% in human proofs.

This review systematically surveys the progress, challenges, and prospects of AI for Mathematics (AI4Math), organizing the field into problem-specific modeling (guiding intuition, constructing counterexamples, formal reasoning in closed systems) and general-purpose modeling (natural language reasoning, formal reasoning, mathematical information retrieval), and reports the authors' own PKU undergraduate and PhD qualifying exam evaluations: GPT-4 averages below 60 while reasoning-enhanced models such as o1, DeepSeek-R1, o3-mini, and Gemini 2.5 Pro mostly exceed 90, with o3-mini averaging 84.4 on 58 PhD qualifying exam problems, while research-level mathematics remains an open challenge.

This review systematically surveys the progress, challenges, and prospects of AI for Mathematics (AI4Math), organizing the field into problem-specific modeling (guiding intuition, constructing counterexamples, formal reasoning in closed systems) and general-purpose modeling (natural language reasoning, formal reasoning, mathematical information retrieval), and reports the authors' own PKU undergraduate and PhD qualifying exam evaluations: GPT-4 averages below 60 while reasoning-enhanced models such as o1, DeepSeek-R1, o3-mini, and Gemini 2.5 Pro mostly exceed 90, with o3-mini averaging 84.4 on 58 PhD qualifying exam problems, while research-level mathematics remains an open challenge.