topic
For 100% of the data, T<=1000, p<=10^7
answer
to catch this god
The general form of Euler's theorem:
\[a^{m} \equiv a^{m \mod \varphi(p) + \varphi(p)} \pmod p\]
We make
\[ans(p) = 2^{2^{2^{...}}} \mod p\]
then
\[ans(p) = 2^{ans(\varphi(p)) + \varphi(p)} \mod p\]
\(O(\log p)\) recursive
#include<iostream>
#include<cstdio>
#include<cmath>
#include<bitset>
#include<cstring>
#include<algorithm>
#define LL long long int
#define Redge(u) for (int k = h[u],to; k; k = ed[k].nxt)
#define REP(i,n) for (int i = 1; i <= (n); i++)
#define BUG(s,n) for (int i = 1; i <= (n); i++) cout<<s[i]<<' '; puts("");
using namespace std;
const int maxn = 10000005,maxm = 100005,INF = 1000000000;
inline int read(){
int out = 0,flag = 1; char c = getchar();
while (c < 48 || c > 57){if (c == '-') flag = -1; c = getchar();}
while (c >= 48 && c <= 57){out = (out << 3) + (out << 1) + c - 48; c = getchar();}
return out * flag;
}
bitset<maxn> isn;
int p[maxn],phi[maxn],pi;
void init(){
phi[1] = 1;
for (int i = 2; i <= 10000000; i++){
if (!isn[i]) p[++pi] = i,phi[i] = i - 1;
for (int j = 1; j <= pi && i * p[j] <= 10000000; j++){
isn[i * p[j]] = true;
if (i % p[j] == 0){
phi[i * p[j]] = phi[i] * p[j];
break;
}
phi[i * p[j]] = phi[i] * (p[j] - 1);
}
}
}
int qpow(int a,int b,int p){
int ans = 1;
for (; b; b >>= 1,a = 1ll * a * a % p)
if (b & 1) ans = 1ll * ans * a % p;
return ans;
}
int Ans(int p){
if (p == 1) return 0;
return qpow(2,Ans(phi[p]) + phi[p],p);
}
int main(){
init();
int T = read(),p;
while (T--){
p = read();
printf("%d\n",Ans(p));
}
return 0;
}