人类遗忘的难题解法，被GPT-5重新找出来了

Core Insights - The article discusses the resolution of a mathematical problem known as Erdős Problem 339, which was previously marked as "unsolved" but was actually resolved in 2003 [2][11]. - The discovery was made using GPT-5 Pro, which identified the relevant literature through an image of the problem [4][13]. - The problem pertains to additive bases in number theory and questions the density of integers that can be expressed as the sum of distinct elements from a set [6][11]. Group 1: Problem Details - Erdős Problem 339 is a classic problem in additive number theory, asking whether the set of integers that can be expressed as the sum of exactly r distinct elements from a set A has positive lower density [6]. - A related question posed by Erdős and Graham concerns the upper density of such sums if the lower density is positive [7]. Group 2: Community Engagement - Prior to the revelation by GPT-5 Pro, discussions among mathematicians on the problem were ongoing, with various interpretations and attempts to construct counterexamples [9][10]. - Notable contributions included references to Waring's Problem, which sparked debates about the implications for Erdős Problem 339 [8]. Group 3: Historical Context - The resolution of Erdős Problem 339 was published in a paper by Hegyvari, Hennecart, and Plagne in 2003, which provided a proof for the conjecture [11][12]. - Paul Erdős, the mathematician behind the problem, is recognized for his extensive contributions to various fields of mathematics, including number theory and combinatorial mathematics [14][18].