[ACM] POJ 2689 Prime Distance (大区间素数筛选）

Prime Distance

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 12811		Accepted: 3420

Description

The branch of mathematics called number theory is about properties of numbers. One of the areas that has captured the interest of number theoreticians for thousands of years is the question of primality. A prime number is a number that is has no proper factors (it is only evenly divisible by 1 and itself). The first prime numbers are 2,3,5,7 but they quickly become less frequent. One of the interesting questions is how dense they are in various ranges. Adjacent primes are two numbers that are both primes, but there are no other prime numbers between the adjacent primes. For example, 2,3 are the only adjacent primes that are also adjacent numbers. 
Your program is given 2 numbers: L and U (1<=L< U<=2,147,483,647), and you are to find the two adjacent primes C1 and C2 (L<=C1< C2<=U) that are closest (i.e. C2-C1 is the minimum). If there are other pairs that are the same distance apart, use the first pair. You are also to find the two adjacent primes D1 and D2 (L<=D1< D2<=U) where D1 and D2 are as distant from each other as possible (again choosing the first pair if there is a tie).

Input

Each line of input will contain two positive integers, L and U, with L < U. The difference between L and U will not exceed 1,000,000.

Output

For each L and U, the output will either be the statement that there are no adjacent primes (because there are less than two primes between the two given numbers) or a line giving the two pairs of adjacent primes.

Sample Input

2 17
14 17

Sample Output

2,3 are closest, 7,11 are most distant.
There are no adjacent primes.

Source

Waterloo local 1998.10.17

解题思路：

给出一个区间[L,R]， 范围为1<=L< R<=2147483647，区间长度长度不超过1000000

求距离最近和最远的两个素数（也就是相邻的差最小和最大的素数）

筛两次，第一次筛出1到1000000的素数，因为1000000^2已经超出int范围，这样的素数足够了。

函数getPrim()；   prime[ ] 存第一次筛出的素数，总个数为prime[0] 

第二次利用已经筛出的素数去筛L,R之间的素数

函数getPrime2();     isprime[] 判断该数是否为素数 prime2[ ]筛出的素数有哪些，一共有prime2[0]个

代码：

#include <iostream>
#include <string.h>
#include <stdio.h>
#include <cmath>
#include <algorithm>
using namespace std;

const int maxn=1e6;
int prime[maxn+10];

void getPrime()
{
    memset(prime,0,sizeof(prime));//一开始prime都设为0代表都是素数（反向思考）
    for(int i=2;i<=maxn;i++)
    {
        if(!prime[i])
            prime[++prime[0]]=i;
        for(int j=1;j<=prime[0]&&prime[j]<=maxn/i;j++)
        {
            prime[prime[j]*i]=1;//prime[k]=1；k不是素数
            if(i%prime[j]==0)
                break;
        }
    }
}

bool isprime[maxn+10];
int prime2[maxn+10];

void getPrime2(int L,int R)
{
    memset(isprime,1,sizeof(isprime));
    //isprime[0]=isprime[1]=0;//这句话不能加，考虑到左区间为2的时候，加上这一句，素数2,3会被判成合数
    if(L<2) L=2;
    for(int i=1;i<=prime[0]&&(long long)prime[i]*prime[i]<=R;i++)
    {
        int s=L/prime[i]+(L%prime[i]>0);//计算第一个比L大且能被prime[i]整除的数是prime[i]的几倍,从此处开始筛
        if(s==1)//很特殊,如果从1开始筛的话，那么2会被筛成非素数
            s=2;
        for(int j=s;(long long)j*prime[i]<=R;j++)
            if((long long)j*prime[i]>=L)
            isprime[j*prime[i]-L]=false; //区间映射 ，比如区间长度为4的区间[4,7],映射到[0,3]中，因为题目范围2,147,483,647数组开不出来
    }
    prime2[0]=0;
    for(int i=0;i<=R-L;i++)
        if(isprime[i])
        prime2[++prime2[0]]=i+L;
}

int main()
{
    getPrime();
    int L,R;
    while(scanf("%d%d",&L,&R)!=EOF)
    {
        getPrime2(L,R);
        if(prime2[0]<2)
            printf("There are no adjacent primes.\n");
        else
        {
            int x1=0,x2=1000000,y1=0,y2=0;
            for(int i=1;i<prime2[0];i++)
            {
                if(prime2[i+1]-prime2[i]<x2-x1)
                {
                    x1=prime2[i];
                    x2=prime2[i+1];
                }
                if(prime2[i+1]-prime2[i]>y2-y1)
                {
                    y1=prime2[i];
                    y2=prime2[i+1];
                }
            }
            printf("%d,%d are closest, %d,%d are most distant.\n",x1,x2,y1,y2);
        }
    }
    return 0;
}