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[数论] 欧拉函数与平方因子

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青青子衿 posted 2019-8-4 17:30 |Read mode
正整数M与其欧拉函数φ(M)互素,是不是M没有平方因子?
Select[Select[Range[1000000], GCD[#, EulerPhi[#]] == 1 &], Not@*SquareFreeQ]

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hbghlyj posted 2019-8-6 08:16
Last edited by hbghlyj 2023-5-15 12:21逆否命题:$N$有平方因子$⇒N$与$φ(N)$有>1公因子
$p∣φ(p^2)=p(p-1)$
$p^2∣N⇒φ(p^2)∣φ(N)⇒p∣φ(N)$

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hbghlyj posted 2023-5-15 19:20
Last edited by hbghlyj 2023-5-16 08:10
hbghlyj 发表于 2019-8-6 01:19
逆命题
$N$与$φ(N)$有>1公因子$⇒N$有平方因子
逆命题不成立。见下方

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$\varphi(10)=4$.  posted 2023-5-16 14:54

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