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格林-陶定理.
格林-陶定理(英语:Green-Tao theorem)是本·格林和陶哲轩于2004年证明的一个关于质数组成的等差数列存在性定理[1]。质数序列包含任意长的等差数列,是格林-陶定理的著名推论。
推论
格林-陶定理有以下两个直接的推论:
- 对于任意正整数
,质数序列中存在任意多长度为
的等差子序列
- 质数序列中包含有任意长的等差子序列
目前已知的最长质数等差数列
质数序列中长度为
的等差子序列,对于1≤n≤k,目前最好的结果是对于k=26,此等差数列为:
- {an=43,142,746,595,714,191 + 23,681,770 · 223,092,870 ·(n-1)}
相关定理与猜想
- 格林-陶定理是塞迈雷迪定理在素数集上的推广。
- 格林-陶定理是埃尔德什等差数列猜想的一个特例。
- 更强的猜想是对于任何正整数r,质数序列中都存在任意长度非r−1阶阶差数列的r阶阶差数列(0阶阶差数列是常数数列,1阶阶差数列是等差数列,依此类推),格林-陶定理就是r=1的特例。对于2阶阶差数列,质数序列中长度为
的二阶阶差子序列,对于0≤n≤k−1,目前最好的结果是对于k=45,此数列为36n^2-810n+2753(不管各项的大小顺序,只要序列中没有重复的质数就可以)。
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