Rabin Miller prime test

This paper does not prove the correct probability , Because I won't
Determine whether large integers are prime numbers

Premise theorem

1. Fermat's small Theorem ： if $n$ Prime number ,$a<n$, be $a^{n-1}\equiv 1\ (mod\ n)$; If any integer $x$ Satisfy $a^{x-1}\equiv 1\ (mod\ x)$, be $x$ There is a high probability that it is a prime number ;
2. Theorem 2： if $p$ For more than 2 The prime number of , be $a^2\equiv 1\ (mod\ p)$ The solution of can only be $a\equiv 1$ or $a\equiv -1$;

Rabin-Miller Prime number test

If the number to be measured $n$ For more than 2 The prime number of , be $n-1$ For the even
set up $n-1=2^s\times d$
Random one less than $n$ The integer of $a$
$a^{2^s\times d}\equiv1\ (mod\ n)$
${(a^{2^{s-1}\times d})}^2\equiv1\ (mod\ n)$
$a^{2^{s-1}\times d}\equiv1\ (mod\ n)$ or $a^{2^{s-1}\times d}\equiv-1\ (mod\ n)$
if 1, You can also continue recursion , Until get -1 Or get $a^d\equiv ±1\ (mod\ n)$

The implementation , Turn it upside down
First, calculate $a^d$, Judge whether it is $±1$
And then keep squaring , obtain $a^{2\times d},a^{2^2\times d},a^{2^3\times d}...a^{2^s\times d}$
Until one step in the middle gets $-1$, Pass the test .
If you get -1 Before , Got it 1, It means that the number is not equal to ±1 The square of the number of gets 1, Not satisfied with the theorem 2, Then the number test fails

Generally, you have to pick a lot at random $a$, Test many times to improve the probability .

But in OI in , Use this string a That's it
$\{2, 3, 5, 7, 11, 13, 17, 19, 23\}$
It guarantees $3,825,123,056,546,413,051$ The number within is calculated correctly （ Has more than long long）

Code

Edition title ：HDU2138

#include<cstdio>
const int TEST_VAL[]={
   2,3,5,7,11,13,17,19,23,29,31,37,41},TEST_NUM=13;

long long PowMod(long long a,long long b,long long p)
{
    long long res=1LL;
    while(b)
    {
        if(b&1LL)
            res=1LL*res*a%p;
        a=1LL*a*a%p;
        b>>=1LL;
    }
    return res;
}

bool MillerRabin(long long a,long long n,long long d)
{
    long long x=PowMod(a,d,n);
    if(x==1||x==n-1)
        return true;
    while(d<n-1)
    {
        x=1LL*x*x%n;
        d=d*2LL;
        if(x==1)
            return false;
        if(x==n-1)
            return true;
    }
    return false;
}
bool isPrime(long long x)
{
    if(x<10)
    {
        if(x==2||x==3||x==5||x==7)
            return true;
        return false;
    }
    long long d=x-1;
    while((d&1LL)==0)
        d>>=1LL;
    for(int i=0;i<TEST_NUM;i++)
        if(TEST_VAL[i]<x&&!MillerRabin(TEST_VAL[i],x,d))
            return false;
    return true;
}

int main()
{
    int n,ans;
    long long x;
    while(scanf("%d",&n)!=EOF)
    {
        ans=0;
        while(n--)
        {
            scanf("%I64d",&x);
            ans+=isPrime(x);
        }
        printf("%d\n",ans);
    }

    return 0;
}