COJ0700 Mathematics (I.)

Question Description

Now there are a lot of numbers, please test these numbers.

Input

First line: CAs, which represents the number of data groups (not greater than 500000), the following CAS lines, one number per line, guaranteed to be within the 64-bit long shaping range, and no negative numbers. You need to test for each number to see if it is prime.

Output

A total of one row, the number of output primes, guaranteed to be positive in the 64-bit long shaping range.

Input example

4 2 3) 5 4

Output example

3

Other Notes

Data range: Guaranteed cas<=100000 to ensure that all numbers are within the 64-bit long range. Please try to optimize your program, including Oi optimization. Welcome to the violence law. →→

First there is such a theorem:

If P is a prime number, X is an integer, and x^2 mod p = 1, then x≡±1 (mod p).

Proof: x^2 mod p = 1 p | X^2-1-P | (x-1) (x + 1), and because P is a prime number, there is a multiple of p in X-1 and x+1.

But the converse of this theorem is a false proposition, but it rarely has a counter-example. We can use Converse to judge the primes.

Set the number to be measured as n, whichever is smaller than n a positive integer A, set n-1 = R * 2^s, if n is prime, the following conditions at least one is established:
1.a^s mod n = 1

2. There is an integer I satisfies: 0<=i<s and a^ (d* (2^i)) mod n = 1

Repeat steps 3 to 4 to stabilize the solution.

#include <cstdio>#include<cctype>#include<cstring>#include<algorithm>#defineRep (s,t) for (int i=s;i<=t;i++)using namespaceStd;inlineintRead () {intx=0, f=1;CharC=GetChar ();  for(;! IsDigit (c); C=getchar ())if(c=='-') f=-1;  for(; IsDigit (c); C=getchar ()) x=x*Ten+c-'0'; returnx*F;} typedefLong LongLl;ll Pow (ll n,ll m,ll p) {ll ans=1, t=N;  while(m) {if(m&1) (ans*=t)%=p; (T*=T)%=p;m>>=1; }    returnans;}intCheck (ll a,ll n,ll R,ll s) {ll ans=pow (A,r,n);if(ans==1)return 0; Rep (1, s) {        if(ans==n-1)return 0; (Ans*=ans)%=N; }    return 1;}intIsPrime (ll N) {if(n==2)return 1; if(n<=1||! (n&1))return 0; intr=n-1, s=0;  while(! (r&1)) r>>=1, s++; Rep (1,3)if(Check (rand ()% (n1)+1, n,r,s))return 0; return 1;}intMain () {intT=read (), ans=0;  while(t--) ans+=IsPrime (read ()); printf ("%d\n", ans); return 0;}

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COJ0700 Mathematics (I.)