2013
11-13

# Pseudoprime numbers

Fermat’s theorem states that for any prime number p and for any integer a > 1, ap = a (mod p). That is, if we raise a to the pth power and divide by p, the remainder is a. Some (but not very many) non-prime values of p, known as base-a pseudoprimes, have this property for some a. (And some, known as Carmichael Numbers, are base-a pseudoprimes for all a.)

Given 2 < p ≤ 1000000000 and 1 < a < p, determine whether or not p is a base-a pseudoprime.

Input contains several test cases followed by a line containing "0 0". Each test case consists of a line containing p and a.

For each test case, output "yes" if p is a base-a pseudoprime; otherwise output "no".

3 2
10 3
341 2
341 3
1105 2
1105 3
0 0


no
no
yes
no
yes
yes


//* @author: 82638882@163.com
import java.util.Scanner;
public class Main
{
public static void main(String[] args)
{
Scanner in=new Scanner(System.in);
while(true)
{
long p=in.nextLong();
long a=in.nextLong();
if(a==0&&p==0) break;
if(isPrime(p)){
System.out.println("no");
continue;
}
System.out.println(sum(a,p,p)==a?"yes":"no");
}
}

public static boolean isPrime(long a)
{
if(a==2) return true;
for(int i=3;i<=Math.sqrt(a);i+=2)
if(a%i==0) return false;
return true;
}

public static long sum(long a,long n,long p)
{
if(n==0) return 1;
long w=sum(a*a%p,n/2,p);
if(n%2==1) w=(w*a)%p;
return w;
}
}