Euler函數(shù):
m = p1^r1 * p2^r2 * …… * pn^rn ai >= 1 , 1 <= i <= n
Euler函數(shù)：
定義：phi(m) 表示小于等于m并且與m互質(zhì)的正整數(shù)的個數(shù)。
phi(m) = p1^(r1-1)*(p1-1) * p2^(r2-1)*(p2-1) * …… * pn^(rn-1)*(pn-1)
= m*(1 - 1/p1)*(1 - 1/p2)*……*(1 - 1/pn)
= p1^(r1-1)*p2^(r2-1)* …… * pn^(rn-1)*phi(p1*p2*……*pn)
定理：若(a , m) = 1 則有 a^phi(m) = 1 (mod m) 即a^phi(m) - 1 整出m
在實(shí)際代碼中可以用類似素數(shù)篩法求出
for (i = 1 i < MAXN i++)
phi[i] = i
for (i = 2 i < MAXN i++)
if (phi[i] == i)
{
for (j = i j < MAXN j += i)
{
phi[j] /= i
phi[j] *= i - 1
}
}
容斥原理：定義phi(p) 為比p小的與p互素的數(shù)的個數(shù)
設(shè)n的素因子有p1, p2, p3, … pk
包含p1, p2…的個數(shù)為n/p1, n/p2…
包含p1*p2, p2*p3…的個數(shù)為n/(p1*p2)…
phi(n) = n - sigm_[i = 1](n/pi) + sigm_[i!=j](n/(pi*pj)) - …… +- n/(p1*p2……pk)
= n*(1 - 1/p1)*(1 - 1/p2)*……*(1 - 1/pk)
標(biāo)簽：
Euler
lt
phi
函數(shù)
上傳時間：
2014-01-10
上傳用戶：wkchong
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標(biāo)簽：
technologies
and
standards
Edition
上傳時間：
2014-01-05
上傳用戶：hj_18
