📄 中文摘要
利用Gemini模型对Bloom的Erdős问题数据库中标记为“开放”的700个猜想进行系统性评估，展示了半自主数学发现的案例研究。采用混合方法，首先通过AI驱动的自然语言验证技术缩小搜索空间，随后由人类专家进行正确性和新颖性评估。针对数据库中标记为“开放”的13个问题，其中5个通过看似新颖的自主解决方案得到解决。此方法旨在结合大型语言模型（LLM）的模式识别与推理能力，与人类数学家的专业知识和直觉，以加速数学发现过程。Gemini在此过程中不仅协助筛选和初步验证猜想，还能在一定程度上生成潜在的证明思路或反例。通过AI的初步筛选，大大降低了人类专家需要处理的问题数量，提高了效率。
📄 English Summary
Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erd\H{o}s Problems
A case study in semi-autonomous mathematics discovery is presented, utilizing the Gemini model to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erdős Problems database. The methodology employs a hybrid approach: initially, AI-driven natural language verification narrows the search space, followed by human expert evaluation to assess correctness and novelty. Thirteen problems marked 'Open' in the database are addressed, with five resolved through seemingly novel autonomous solutions. This approach aims to combine the pattern recognition and reasoning capabilities of large language models (LLMs) with the expertise and intuition of human mathematicians, thereby accelerating the mathematical discovery process. Gemini not only assists in filtering and preliminary verification of conjectures but also generates potential proof ideas or counterexamples to some extent. The AI's initial screening significantly reduces the number of problems human experts need to address, enhancing efficiency. Human experts are then responsible for rigorously reviewing AI-proposed solutions, ensuring their mathematical rigor, correctness, and evaluating their novelty within existing mathematical literature. The case study indicates that AI offers significant advantages in handling complex mathematical problems, particularly in identifying potential connections and simplifying expressions. However, for highly abstract problems requiring deep conceptual understanding, human insight remains indispensable. This human-AI collaborative model offers an efficient and promising paradigm for future mathematical research, especially when dealing with large sets of unsolved problems. The findings underscore AI's immense potential in assisting mathematicians to explore unknown domains and accelerate scientific discovery, laying the groundwork for developing more intelligent and autonomous mathematical discovery tools.