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数学研究生解决加法极限问题

2025-05-23 19:14 by 少女骑士变身记

加法是数学中最简单概念之一,但数学家对于加法所产生的各种模式还有很多尚未解答的疑问。其中一个问题与无和集(Sum-free set)的性质有关。所谓无和集是指该集合中任意取三个数两数相加不会等于第三个数,举例来说,奇数集合中任意两数相加是偶数,奇数集显然是无和集。1965 年著名数学家 Paul Erdős 提出了一个简单问题:无和集有多普遍?这个问题过去几十年进展甚微。在该问题提出 60 年之后,牛津大学研究生 Benjamin Bedert 终于给出了证明:任何整数集合,必定有一个大子集是无和集。随机选择一百万个整数,其中半数是奇数,奇数集是无和集,那么该集合的一个无和子集就包含有大约 50 万个数。Erdos 在 1965 年的论文中仅用几行就推导出了一个下界:N/3——任意包含 N 个整数的集合,至少有一个包含 N/3 个整数的无和子集。最大的无和子集肯定会超过 N/3,但多大?Bedert 的结论是 N/3 + log(log N)。

www.quantamagazine.org/graduate-student-solves-classic-problem-about-the-limits-of-addition-20250522/
en.wikipedia.org/wiki/Sum-free_set

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数学研究生解决加法极限问题

2025-05-23 19:14 by 少女骑士变身记

加法是数学中最简单概念之一,但数学家对于加法所产生的各种模式还有很多尚未解答的疑问。其中一个问题与无和集(Sum-free set)的性质有关。所谓无和集是指该集合中任意取三个数两数相加不会等于第三个数,举例来说,奇数集合中任意两数相加是偶数,奇数集显然是无和集。1965 年著名数学家 Paul Erdős 提出了一个简单问题:无和集有多普遍?这个问题过去几十年进展甚微。在该问题提出 60 年之后,牛津大学研究生 Benjamin Bedert 终于给出了证明:任何整数集合,必定有一个大子集是无和集。随机选择一百万个整数,其中半数是奇数,奇数集是无和集,那么该集合的一个无和子集就包含有大约 50 万个数。Erdos 在 1965 年的论文中仅用几行就推导出了一个下界:N/3——任意包含 N 个整数的集合,至少有一个包含 N/3 个整数的无和子集。最大的无和子集肯定会超过 N/3,但多大?Bedert 的结论是 N/3 + log(log N)。

www.quantamagazine.org/graduate-student-solves-classic-problem-about-the-limits-of-addition-20250522/
en.wikipedia.org/wiki/Sum-free_set

#数学

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