AlphaEvolve is a novel evolutionary coding agent that integrates the generative power of large language models (LLMs) with automated evaluation within an iterative, evolutionary framework. Designed to autonomously tackle complex scientific and practical problems, AlphaEvolve proposes algorithmic solutions, tests them through rigorous evaluation, and iteratively refines them to improve performance. In this work, we present AlphaEvolve as a transformative tool for mathematical discovery, capable of exploring vast solution spaces to identify novel constructions and advance understanding of longstanding open problems. To demonstrate its capabilities, we applied AlphaEvolve to a diverse set of 67 mathematical challenges drawn from fields including analysis, combinatorics, geometry, and number theory. In the majority of cases, the system successfully rediscovered the best-known solutions. More significantly, it discovered improved solutions in several instances, achieving results that surpass previously published outcomes. In some cases, AlphaEvolve went beyond mere optimization by generalizing patterns observed for finite inputs into closed-form mathematical expressions valid for all inputs—showcasing its ability to uncover deeper structural insights. The framework also demonstrates strong synergy when combined with other advanced AI systems. By integrating AlphaEvolve with Deep Think for enhanced reasoning and AlphaProof for automated theorem proving, we created a comprehensive pipeline that not only generates candidate solutions but also verifies their correctness and provides formal mathematical justifications. This multi-agent approach enables a self-contained loop of hypothesis generation, testing, refinement, and proof, significantly reducing the need for manual intervention. These results highlight the potential of LLM-guided evolutionary search to complement and extend human mathematical intuition. AlphaEvolve consistently delivers high-quality results with minimal setup and reduced computational overhead compared to traditional methods, making it a scalable tool for tackling complex optimization and discovery tasks. The system’s ability to operate at scale, adaptively explore solution spaces, and produce verifiable outcomes positions it as a powerful new paradigm in mathematical research. We present AlphaEvolve not as a replacement for human mathematicians, but as a collaborative partner capable of accelerating discovery, uncovering hidden patterns, and offering new perspectives on difficult problems. This work marks a significant step toward a future where AI systems play an active, creative role in scientific advancement, particularly in domains where human exploration is limited by time, complexity, or cognitive load.