super_log （广义欧拉降幂）（2019南京网络赛）
作者：互联网
题目：
In Complexity theory, some functions are nearly O(1)O(1), but it is greater then O(1)O(1). For example, the complexity of a typical disjoint set is O(nα(n))O(nα(n)). Here α(n)α(n) is Inverse Ackermann Function, which growth speed is very slow. So in practical application, we often assume α(n) \le 4α(n)≤4.
However O(α(n))O(α(n)) is greater than O(1)O(1), that means if nn is large enough, α(n)α(n) can greater than any constant value.
Now your task is let another slowly function log*log∗ xx reach a constant value bb. Here log*log∗ is iterated logarithm function, it means “the number of times the logarithm function iteratively applied on xx before the result is less than logarithm base aa”.
Formally, consider a iterated logarithm function log_{a}^*loga∗
Find the minimum positive integer argument xx, let log_{a}^* (x) \ge bloga∗(x)≥b. The answer may be very large, so just print the result xx after mod mm.
Input
The first line of the input is a single integer T(T\le 300)T(T≤300) indicating the number of test cases.
Each of the following lines contains 33 integers aa, bb and mm.
1 \le a \le 10000001≤a≤1000000
0 \le b \le 10000000≤b≤1000000
1 \le m \le 10000001≤m≤1000000
Note that if a==1, we consider the minimum number x is 1.
Output
For each test case, output xx mod mm in a single line.
Hint
In the 4-th4−th query, a=3a=3 and b=2b=2. Then log_{3}^* (27) = 1+ log_{3}^* (3) = 2 + log_{3}^* (1)=3+(-1)=2 \ge blog3∗(27)=1+log3∗(3)=2+log3∗(1)=3+(−1)=2≥b, so the output is 2727 mod 16 = 1116=11.
样例输入
5 2 0 3 3 1 2 3 1 100 3 2 16 5 3 233
样例输出
1 1 3 11 223
解题报告：严重怀疑这场比赛的出题方，读题实在是太困难了，看了半个小时才理解了要求解什么东西，一看就是一个广义欧拉降幂，然后就直接去写的之前的模板，求解无穷个2的幂次，
但是修修改改的过了样例，但是忘记了广义欧拉降幂的实现，疯狂wa，在比赛结束后找了好久才发现是没有去使用广义欧拉降幂的性质，忘记在该加欧拉（m）的地方进行加了，隔壁队的
数论手，给了一个能够实现这个功能的快速幂，稍微一修改就A了。这个需要掌握了，当时时间太紧，头脑已经混了。
ac代码：
1 #include<iostream>
2 #include<cstring>
3 #include<cstdio>
4 #include<cmath>
5 #include<algorithm>
6 #include<map>
7 #include<vector>
8 using namespace std;
9 typedef long long ll;
10
11 const int maxn=1e7+10;
12 vector<int> prime;
13 int phi[maxn];
14
15 void db_Onion()
16 {
17 phi[1]=1;
18 for(int i=2;i<maxn;i++)
19 {
20 if(phi[i]==0)
21 {
22 prime.push_back(i);
23 phi[i]=i-1;
24 }
25 for(int j=0;j<prime.size()&&prime[j]*i<maxn;j++)
26 {
27 if(i%prime[j]==0)
28 {
29 phi[i*prime[j]]=phi[i]*prime[j];
30 break;
31 }
32 else
33 {
34 phi[i*prime[j]]=phi[i]*(prime[j]-1);
35 }
36 }
37 }
38 }
39 ll a,b,m;
40 ll cal(ll x,ll m)
41 {
42 if(x<m)
43 return x;
44 else
45 return x%m+m;
46 }
47
48 ll ans;
49
50 ll ksm(ll r,ll n,ll m)
51 {
52 ll d=r,ans=1;
53 bool f=0;
54 if(r==0&&n==0)return 1;
55 while(n){
56 if(n&1){
57 ans=ans*d;
58 if(ans>=m){
59 ans%=m;f=1;
60 }
61 }
62 d=d*d;
63 if(n>1&&d>=m){
64 d%=m;f=1;
65 }
66 n>>=1;
67 }
68 if(f)ans+=m;
69 return ans;
70 }
71 ll slove(ll p,int tot)
72 {
73 if(tot==b+1)
74 return 1;
75 if(p==1)
76 return 1;
77
78 ll tmp=slove(phi[p],tot+1);
79 return ksm(a,tmp,p);
80 }
81
82 int main()
83 {
84 db_Onion();
85 int t;
86 scanf("%d",&t);
87 while(t--)
88 {
89 scanf("%lld%lld%lld",&a,&b,&m);
90 if(b==0||a==1)
91 {
92 printf("%lld\n",1%m);
93 continue;
94 }
95 ans=0;
96 printf("%lld\n",slove(m,1)%m);
97 }
98 }
标签：le,log,降幂,int,ll,xx,2019,include 来源： https://www.cnblogs.com/Spring-Onion/p/11444096.html