Researchers Break New Ground in Sum and Difference of Sets Problem, Surpassing AlphaEvolve's Bound

On May 14, 2025, DeepMind announced significant advancements in solving mathematical problems using their large language model, AlphaEvolve. Among these achievements, AlphaEvolve notably improved the known lower bound for the sum and difference of sets problem. By applying a set containing 54,265 integers, AlphaEvolve raised the lower bound from θ = 1.14465 to θ = 1.1584. This progress marked a substantial leap in the field, showcasing the potential of AI in tackling complex mathematical challenges. Building on this foundation, this paper presents an even more refined solution. Our team has constructed an explicit U set—specifically designed to address the sum and difference of sets problem—that contains more than (10^{43546}) elements. This new construction allows us to push the lower bound to θ = 1.173050, a significant improvement over the previous results achieved by AlphaEvolve. To facilitate the extensive computational requirements of this task, we utilized the GNU Multiple Precision Arithmetic Library (GMP), a free software library renowned for its efficiency in handling large numbers. The GMP library provided the necessary support for both integer and floating-point arithmetic, enabling us to manage the vast number of elements in our U set accurately and efficiently. This work not only enhances the understanding of the sum and difference of sets problem but also highlights the ongoing competition and collaboration between human mathematicians and AI systems in advancing mathematical knowledge. The significant improvement in the lower bound underscores the potential of combining traditional mathematical techniques with advanced computational tools. Future research could further explore the capabilities of AI in mathematical problem-solving and the development of even more optimized constructions. The findings presented here contribute to the broader conversation on the role of AI in mathematics and its potential to unlock new insights and solutions in areas previously thought intractable.