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[数论] 两两互素的反例

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hbghlyj posted 2019-7-24 17:03 |Read mode
整数$a_i$(i=1,2…n!)每n+1个构成的数组都互素,那么整数$a_i$是否两两互素

n=3时反例如下.取两两互素整数$p_1,p_2,p_3,p_4$
那么,$p_1p_2,p_1p_3,p_1p_4,p_2p_3,p_2p_4,p_3p_4$六个数每4个构成的数组都互素,然而其中有15-3=12对整数不互素
n=4时???

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realnumber posted 2019-7-25 08:24
就按你的来,n=4,p1,p2,p3,p4,p5
p1p2,p1p3,...10个中任意取5个,最大公因素是1.

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original poster hbghlyj posted 2019-7-25 11:45
回复 2# realnumber 您看错题了。题目是n!个整数。所以n=4应该是24个整数。

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realnumber posted 2019-7-25 11:51
回复 3# hbghlyj

哦,是看错了,不过可以补完整,再添上不同与p1,p2,p3,p4,p5,的一些素数,一直到24个.

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original poster hbghlyj posted 2019-7-25 14:54
Last edited by hbghlyj 2019-7-25 15:11回复 4# realnumber 既然这个构造可以推广到任何不小于3的正整数n,我们继续挖掘,修改一下条件:
设正整数2<n<m,整数$a_i$(i=1,2…m)每n个构成的数组都不互素,那么整数$a_i$每n+1个能否都互素

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realnumber posted 2019-7-26 08:35
还是沿用上面办法
初始$a_i$=1,i=1,2,3,...,m
设正整数2<n<m,每n个构成的数组都乘以一个素数$p_t,t=1,2,3,..C_m^n$,那么整数a i  ai每n+1个都互素

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