ZOJ 3483 Gaussian Prime(数学啊 )

题目链接：http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemId=4280

In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The prime elements of Z[i] are also known as Gaussian primes. Gaussian integers can be uniquely factored in terms of Gaussian primes up to powers of i and rearrangements.

A Gaussian integer a + bi is a Gaussian prime if and only if either:

• One of a, b is zero and the other is a prime number of the form 4n + 3 (with n a nonnegative integer) or its negative -(4n + 3), or
• Both are nonzero and a² + b² is a prime number (which will not be of the form 4n + 3).

0 is not Gaussian prime. 1, -1, i, and -i are the units of Z[i], but not Gaussian primes. 3, 7, 11, ... are both primes and Gaussian primes. 2 is prime, but is not Gaussian prime, as 2 = i(1-i)².

Your task is to calculate the density of Gaussian primes in the complex plane [x1, x2] × [y1, y2]. The density is defined as the number of Gaussian primes divided by the number of Gaussian integers.

Input

There are multiple test cases. The first line of input is an integer T ≈ 100 indicating the number of test cases.

Each test case consists of a line containing 4 integers -100 ≤ x1 ≤ x2 ≤ 100, -100 ≤ y1 ≤ y2 ≤ 100.

Output

For each test case, output the answer as an irreducible fraction.

Sample Input

30 0 0 00 0 0 100 3 0 3

Sample Output

0/12/117/16

References

• http://en.wikipedia.org/wiki/Gaussian_integer
• Weisstein, Eric W. "Gaussian Prime." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GaussianPrime.html

代码如下：

//#pragma warning (disable:4786)#include <cstdio>#include <cmath>#include <cstring>#include <string>#include <cstdlib>#include <climits>#include <ctype.h>#include <queue>#include <stack>#include <vector>#include <utility>#include <deque>#include <set>#include <map>#include <iostream>#include <algorithm>using namespace std;const double eps = 1e-9;//const double pi = atan(1.0)*4;const double pi = 3.1415926535897932384626;const double e = exp(1.0);#define INF 0x3f3f3f3f//#define INF 1e18//typedef long long LL;//typedef __int64 LL;#define ONLINE_JUDGE#ifndef ONLINE_JUDGEfreopen("in.txt", "r", stdin);freopen("out.txt", "w", stdout);#endif#define maxn 100000int prim[maxn];void init(){    for(int i = 2; i <= maxn; i++)    {        if(!prim[i])        {            for(int j = i+i; j <= maxn; j+=i)            {                prim[j] = 1;            }        }    }}int GCD(int a, int b){    if(b == 0)        return a;    return GCD(b,a%b);}int main(){    int t;    init();    scanf("%d",&t);    int x1, x2, y1, y2;    while(t--)    {        cin>>x1>>x2>>y1>>y2;        int ans = 0;        int tem;        for(int x = x1; x <= x2; x++)        {            for(int y = y1; y <= y2; y++)            {                if(x == 0)                {                    if(y < 0)                    {                        tem = -y;                    }                    else                        tem = y;                    if((tem-3)%4==0 && prim[tem]==0)                        ans++;                }                else if(y == 0)                {                    if(x < 0)                    {                        tem = -x;                    }                    else                        tem = x;                    if((tem-3)%4==0 && prim[tem]==0)                        ans++;                }                else                {                    tem = x*x+y*y;                    if(prim[tem]==0 && (tem-3)%4!=0)                        ans++;                }            }        }        int tol = (x2-x1+1)*(y2-y1+1);        int gcd = GCD(ans,tol);        printf("%d/%d\n",ans/gcd,tol/gcd);    }    return 0;}