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| #include<bits/stdc++.h>
using namespace std;
#define enter fout<<"\n";
#define space fout<<" ";
#define dot fout<<",";
#define oui fout<<"Yes\n";
#define non fout<<"No\n";
#define si fout<<"?";
#define i32 int
#define u32 unsigned int
#define i64 long long
#define u64 unsigned long long
#define i128 __int128
#define u128 unsigned __int128
#define debug(x) fout<<#x<<"="<<x<<"\n";
#define vdebug(a,n) fout<<#a<<"=[";for(int i=0;i<=n;++i)fout<<a[i]<<" ";fout<<"]\n";
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
namespace fastio{
const int bufl=1<<20;
const double base1[16]={1,1e-1,1e-2,1e-3,1e-4,1e-5,1e-6,1e-7,1e-8,1e-9,1e-10,1e-11,1e-12,1e-13,1e-14,1e-15};
const double base2[16]={1,1e1,1e2,1e3,1e4,1e5,1e6,1e7,1e8,1e9,1e10,1e11,1e12,1e13,1e14,1e15};
struct IN{
FILE *IT;char ibuf[bufl],*is=ibuf,*it=ibuf;
IN(){IT=stdin;}IN(char *a){IT=fopen(a,"r");}
inline char getChar(){if(is==it){it=(is=ibuf)+fread(ibuf,1,bufl,IT);if(is==it)return EOF;}return *is++;}
template<typename Temp>inline void getInt(Temp &a){a=0;int b=0,c=getChar();while(c<48||c>57)b^=(c==45),c=getChar();while(c>=48&&c<=57)a=(a<<1)+(a<<3)+c-48,c=getChar();if(b)a=-a;}
template<typename Temp>inline void getDouble(Temp &a){a=0;int b=0,c=getChar(),d=0;__int128 e=0,f=0;while(c<48||c>57)b^=(c==45),c=getChar();while(c>=48&&c<=57)e=(e<<1)+(e<<3)+c-48,c=getChar();if(c==46){c=getChar();while(c>=48&&c<=57)d++,f=(f<<1)+(f<<3)+c-48,c=getChar();}a=e+base1[d]*f;if(b)a=-a;}
IN& operator>>(char &a){a=getChar();while(a<=32)a=getChar();return *this;}
IN& operator>>(char *a){do{*a=getChar();}while(*a<=32);while(*a>32)*++a=getChar();*a=0;return *this;}
IN& operator>>(string &a){a.clear();char b=getChar();while(b<=32)b=getChar();while(b>32)a+=b,b=getChar();return *this;}
IN& operator>>(int &a){getInt(a);return *this;}
IN& operator>>(long long &a){getInt(a);return *this;}
IN& operator>>(__int128 &a){getInt(a);return *this;}
IN& operator>>(float &a){getDouble(a);return *this;}
IN& operator>>(double &a){getDouble(a);return *this;}
IN& operator>>(long double &a){getDouble(a);return *this;}
};
struct OUT{
FILE *IT;char obuf[bufl],*os=obuf,*ot=obuf+bufl;int Eps;long double Acc;
OUT(){IT=stdout,Eps=6,Acc=0.5;}OUT(char *a){IT=fopen(a,"w"),Eps=6,Acc=0.5;}
inline void ChangEps(int x=6){Eps=x;}
inline void flush(){fwrite(obuf,1,os-obuf,IT);os=obuf;}
inline void putChar(int a){*os++=a;if(os==ot)flush();}
template<typename Temp>inline void putInt(Temp a){if(a<0){putChar(45);a=-a;}if(a<10){putChar(a+48);return;}putInt(a/10);putChar(a%10+48);}
template<typename Temp>inline void putLeading(Temp a,int b){if(!b)return;putLeading(a/10,b-1);putChar(a%10+48);}
template<typename Temp>inline void putDouble(Temp a){if(a<0){putChar(45);a=-a;}__int128 ff=(a-(__int128)a)*base2[Eps+2],gg=0;ff+=50;while(ff>0){ff/=10;gg++;}__int128 b=a;if(gg==Eps+3){putInt(b+1);}else{putInt(b);}a-=b;a*=base2[Eps];b=a+Acc;putChar(46);putLeading(b,Eps);}
OUT& operator<<(char a){putChar(a);return *this;}
OUT& operator<<(const char *a){while(*a)putChar(*a++);return *this;}
OUT& operator<<(string a){for(auto c:a)putChar(c);return *this;}
OUT& operator<<(int a){putInt(a);return *this;}
OUT& operator<<(long long a){putInt(a);return *this;}
OUT& operator<<(__int128 a){putInt(a);return *this;}
OUT& operator<<(unsigned int a){putInt(a);return *this;}
OUT& operator<<(unsigned long long a){putInt(a);return *this;}
OUT& operator<<(unsigned __int128 a){putInt(a);return *this;}
OUT& operator<<(float a){putDouble(a);return *this;}
OUT& operator<<(double a){putDouble(a);return *this;}
OUT& operator<<(long double a){putDouble(a);return *this;}
~OUT(){flush();}
};
}
fastio::IN fin;
fastio::OUT fout;
int vis[1000010];
int p[100010],ptop=0;
short mu[1000010];
int musu[1000010];
int phi[1000010];
long long phisu[1000010];
int d[1000010];
int mnnum[1000010];
void sieve(int n){
phi[1]=1;phisu[1]=1;mu[1]=1;musu[1]=1;d[1]=1;
int tmp;
for(int i=2;i<=n;++i){
if(!vis[i]){
vis[i]=-(++ptop);
p[ptop]=i;
mu[i]=-1;
phi[i]=i-1;
d[i]=2;
mnnum[i]=1;
}
for(int j=1;j<=ptop&&i*p[j]<=n;++j){
vis[i*p[j]]=p[j];
if(i%p[j]==0){
phi[i*p[j]]=phi[i]*p[j];
mnnum[i*p[j]]=mnnum[i]+1;
d[i*p[j]]=d[i]/mnnum[i*p[j]]*(mnnum[i*p[j]]+1);
break;
}else{
mu[i*p[j]]=-mu[i];
phi[i*p[j]]=phi[i]*(p[j]-1);
mnnum[i*p[j]]=1;
d[i*p[j]]=d[i]*2;
}
}
musu[i]=musu[i-1]+mu[i];
phisu[i]=phisu[i-1]+phi[i];
}
}
namespace decompos{
unsigned long long Number(unsigned long long a,unsigned long long b){
mt19937_64 rng(chrono::steady_clock::now().time_since_epoch().count());
return (a+rng()%(b-a+1));
}
unsigned long long stein(unsigned long long x,unsigned long long y){
if(x==y)return x;
unsigned long long a=y-x;
unsigned long long b=x-y;
int n=__builtin_ctzll(b);
unsigned long long s=(x<y?a:b);
unsigned long long t=(x<y?x:y);
return stein(s>>n,t);
}
unsigned long long gcd(unsigned long long x,unsigned long long y){
if(x==0)return y;
if(y==0)return x;
int n=__builtin_ctzll(x);
int m=__builtin_ctzll(y);
return (stein(x>>n,y>>m)<<(n<m?n:m));
}
unsigned long long limit=2147483648;
vector<int>small={2,7,61};
vector<int>large={2,325,9375,28178,450775,9780504,1795265022};
unsigned long long qp(unsigned long long x,unsigned long long y,unsigned long long m){
unsigned long long res=1;
while(y){
if(y&1)res=((unsigned __int128)res*x)%m;
x=((unsigned __int128)x*x)%m;
y>>=1;
}
return res;
}
bool isComposite(unsigned long long a,unsigned long long d,unsigned long long n,unsigned long long s){
unsigned long long x=qp(a,d,n);
if(x==1)return false;
while(s--){
if(x==n-1)return false;
x=((unsigned __int128)x*x)%n;
}
return true;
}
bool isPrime(unsigned long long n){
if(n==1)
return false;
vector<int>bases=(n<limit?small:large);
if(find(bases.begin(),bases.end(),n)!=bases.end())
return true;
for(int p:bases)
if(n%p==0)
return false;
unsigned long long s=__builtin_ctzll(n-1);
unsigned long long d=(n-1)>>s;
for(int base:bases){
if(isComposite(base,d,n,s))
return false;
}
return true;
}
class Montgomery{
unsigned long long m;
unsigned long long r;
public:
Montgomery(unsigned long long n):m(n),r(n){
for(int i=0;i<5;i++){
r*=2-m*r;
}
}
unsigned long long fma(unsigned long long a,unsigned long long b,unsigned long long c)const{
unsigned __int128 d=(unsigned __int128)(a)*b;
unsigned long long e=c+m+(d>>64);
unsigned long long f=(unsigned long long)(d)*r;
unsigned long long g=((unsigned __int128)(f)*m)>>64;
return (e-g);
}
unsigned long long mul(unsigned long long a,unsigned long long b)const{return fma(a,b,0);}
};
unsigned long long PollardRho(unsigned long long n){
if(n%2==0)return 2;
Montgomery m(n);
unsigned long long c1=1;
unsigned long long c2=2;
unsigned long long M=512;
unsigned long long w1=1;
unsigned long long w2=2;
retry:
unsigned long long z1=w1;
unsigned long long z2=w2;
for(unsigned long long k=M;;k <<= 1){
unsigned long long x1=z1+n;
unsigned long long x2=z2+n;
for(unsigned long long j=0;j<k;j+=M){
unsigned long long y1=z1;
unsigned long long y2=z2;
unsigned long long q1=1;
unsigned long long q2=2;
z1=m.fma(z1,z1,c1);
z2=m.fma(z2,z2,c2);
for(unsigned long long i=0;i<M;i++){
unsigned long long t1=x1-z1;
unsigned long long t2=x2-z2;
z1=m.fma(z1,z1,c1);
z2=m.fma(z2,z2,c2);
q1=m.mul(q1,t1);
q2=m.mul(q2,t2);
}
q1=m.mul(q1,x1-z1);
q2=m.mul(q2,x2-z2);
unsigned long long q3=m.mul(q1,q2);
unsigned long long g3=gcd(n,q3);
if(g3==1)continue;
if(g3!=n)return g3;
unsigned long long g1=gcd(n,q1);
unsigned long long g2=gcd(n,q2);
unsigned long long c=(g1!=1?c1:c2);
unsigned long long x=(g1!=1?x1:x2);
unsigned long long z=(g1!=1?y1:y2);
unsigned long long g=(g1!=1?g1:g2);
if(g==n){
do{
z=m.fma(z,z,c);
g=gcd(n,x-z);
}while(g==1);
}
if(g!=n)return g;
w1+=2;
w2+=2;
goto retry;
}
}
}
void factorize(unsigned long long n,vector<unsigned long long>& res){
if(n<=1)return;
if(isPrime(n)){
res.push_back(n);
return;
}
unsigned long long p=PollardRho(n);
factorize(p,res);
factorize(n/p,res);
}
}
long long n,m,res;
int main(){
sieve(1000);
for(i64 x=1;x<=1000;++x){
for(i64 y=1;y<=1000;++y){
i64 upper=x*x*x*x-y*y*y*y;
i64 lower=x*x*x+y*y*y;
if((upper)%(lower)==0&&(upper/lower)>=1&&vis[upper/lower]<=0){
fout<<"! "<<(upper/lower)<<" "<<x<<" "<<y<<"\n";
}
}
}
i64 inf=5e15;
for(i64 x=1;;++x){
i64 tmp=x*x+(x+1)*(x+1);
if(tmp>inf)break;
else{
if(decompos::isPrime(tmp)){
res++;
}
}
if(x%100000==0){
fout<<"! "<<x<<" "<<res<<"\n";
fout.flush();
}
}
fout<<res;
return 0;
}
|
