2017ecjtu-summer training #7 POJ 2689

Posted 2020-09-30 灬从此以后灬

Prime Distance

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 18731		Accepted: 5006

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.

题意：输入区间[l,u]，其中l和u为int范围的整数，区间最大为1000000。求出[l,u]中，相邻素数只差最大和最小的素数对。当存在多个时，输出较小的素数对。

题解：l,u范围太大，不能直接求int范围的素数。而区间间隔比较小，只有1e6，而且对于int范围内的合数来说，最小质因子必定小于2^16。所以可以求出[l,u]中合数，转而求出素数，然后暴力枚举所有素数对即可。

如何求区间[l,u]中的合数：上面已经说了，合数的最小质因子小于2^16，即小于50000。所以先求出小于50000的所有素数。则区间[l,u]中的合数，必定可以表示为小于50000的素数的倍数。对于素数p来说，令a=(l-1)/p+1,b=u/p。则枚举j=a到b，j*p可以枚举所有[l,u]中质因子含有p的合数。枚举所有小于50000的素数，然后用上述方式枚举倍数，即可找出[l,u]中所有的合数。

由于l,u在int范围，所以不能直接用数组标记。需要加个偏移量，取l，则数组大小小于1e6的f[0,u-l]，即可标记。

接着枚举区间中所有的相邻素数对即可。

特别注意：由于1不是小于50000的素数的倍数，所以在与合数相斥中，会被当成素数。需要特别处理下。

 AC代码

 1 #include <stdio.h>
 2 #include <math.h>
 3 #include <string.h>
 4 #include <stdlib.h>
 5 #include <iostream>
 6 #include <sstream>
 7 #include <queue>
 8 #include <vector>
 9 #include <algorithm>
10 #define maxn 50010
11 #define maxm 1000010
12 #define inf  0x3f3f3f3f
13 using namespace std;
14 typedef long long ll;
15 int vis[maxm],f[maxm];
16 int prime[maxn],prime1[maxm];
17 int mind,maxd,minl,minr,maxl,maxr;
18 int t,t1;
19 ll L,R;
20 void init()                          //筛选出50000以内的素数
21 {
22     t=0;
23     memset(vis,0,sizeof(vis));
24     for(int i=2;i<maxn;i++)
25     {
26         if(!vis[i])
27         {
28             prime[t++]=i;
29             for(int j=i+i;j<maxn;j+=i)
30                 vis[j]=1;
31         }
32     }
33 }
34 void selet()
35 {
36     memset(vis,0,sizeof(vis));                   //*特别考虑1
37     for(int i=0;i<t;i++)                         //标记int范围内的合数
38     { 
39         ll b=L/prime[i];
40         while(b*prime[i]<L||b<=1)
41             b++;
42         for(ll j=b*prime[i];j<=R;j+=prime[i])     
43             vis[j-L]=1;                           //节约空间，降低空间复杂度
44     }
45     if(L==1)                                   
46         vis[0]=1;
47     t1=0;
48     for(ll i=L;i<=R;i++)                       //找出区间内的素数存进数组
49     {
50         if(!vis[i-L])
51             prime1[t1++]=i;
52     }
53 }
54 void solve()
55 {
56     selet();
57     mind=inf,maxd=-inf;
58     minl=minr=maxl=maxr=-1;
59     for(int i=1;i<t1;i++)                  //枚举区间内的素数，更新要输出的值
60     {
61         int d=prime1[i]-prime1[i-1];
62         if(d<mind)
63         {
64             mind=d;
65             minl=prime1[i-1];
66             minr=prime1[i];
67         }
68         if(d>maxd)
69         {
70             maxd=d;
71             maxl=prime1[i-1];
72             maxr=prime1[i];
73         }
74     }
75 }
76 int main(int argc, char const *argv[])
77 {
78     init();
79     while(cin>>L>>R)
80     {
81         solve();
82         if(t1<2) 
83             printf("There are no adjacent primes.\n");  
84         else 
85             printf("%d,%d are closest, %d,%d are most distant.\n",minl,minr,maxl,maxr);
86     }
87     return 0;
88 }