POJ-2689 Prime Distance (filtering of Prime numbers in large intervals)

POJ-2689 Prime Distance (filtering of Prime numbers in large intervals)

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 varous ranges. adjacent primes are two numbers that are both primes, but there are no other prime numbers between the adjacent primes. for example, 2 and 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 1714 17

Sample Output

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

Source

Waterloo local 1998.10.17

In the output range [l, r], the two prime numbers are the largest and the smallest.

Idea: Because the range is large and the range is relatively small, we first filter out the prime numbers in the range of 2 ^ 16, then, a small interval is used to delete the Union of a large interval (each union can be obtained by multiplying the prime number ). Last Traversal

#include 
 
  #include 
  
   #include 
   
    #include #include 
    
     typedef long long ll;using namespace std;const int INF = 1<<30;const int inf = 60000;const int maxn = 1000005;int prime[maxn], vis[maxn], cnt;void init() {cnt = 0;for (ll i = 2; i < inf; i++) if (!vis[i]) {prime[cnt++] = i;for (ll j = i*i; j < inf; j += i)vis[j] = 1;}}int isprime[maxn], a[maxn], c;int main() {int l, r;init();while (scanf("%d%d", &l, &r) != EOF) {memset(isprime, 1, sizeof(isprime));if (l == 1) l = 2;for (int i = 0; i < cnt && (ll)prime[i]*prime[i] <= r; i++) {int s = l / prime[i] + (l % prime[i] > 0);if (s == 1) s = 2;for (int j = s; (ll)j*prime[i] <= r; j++) if ((ll)j*prime[i] >= l) isprime[j*prime[i]-l] = 0;}c = 0;for (int i = 0; i <= r-l; i++) if (isprime[i])a[c++] = i + l;if (c < 2) {printf("There are no adjacent primes.\n");continue;}else {int x1 = 0, x2 = 0, y1 = 0, y2 = INF;for (int i = 1; i < c; i++) {if (a[i] - a[i-1] > x2 - x1) {x1 = a[i-1];x2 = a[i];}if (a[i] - a[i-1] < y2 - y1) {y1 = a[i-1];y2 = a[i];}}printf("%d,%d are closest, %d,%d are most distant.\n", y1, y2, x1, x2);}}return 0;}