素数算法
素数，也叫质数，也就是只能被1和其本身整除的整数! 如2，3，5，7，11，13等...
若要求解一个N以内所有的素数，有四种方法：
1. 暴力破解：num从2到sqrt(num)求余，若余数为0则不为素数；
2. 筛选： 先筛除素数2的倍数数，再筛除余下第一个素数3的倍数数, 一直到最后N/2；
3. 静态素数表：建一个大的素数表，查表即可判断(N比较小时)；
4. 6n+1和6n-1法：除了素数2，3外6n+1和6n-1的才可能是素数，但其中有伪素数需剔除；
1. 暴力破解：
#include
#include
#include
#include
#include
#define MAX_NUM 100000
static bool prime(unsigned int num)
{
unsigned int limit = (unsigned int)sqrt(num);
int idx;
for ( idx = 2; idx <= limit; ++idx )
{
if ( 0 == ( num % idx ) ) return false;
}
return true;
}
int main(int argc, char *argv[])
{
int num, count = 0;
struct timeval tmvalBeg, tmvalEnd;
gettimeofday(&tmvalBeg, NULL);
printf("(1, %d) prime is ", MAX_NUM);
for ( num = 2; num <= MAX_NUM; ++num )
{
if ( prime(num) )
{
++count;
printf("%d,", num);
}
}
printf("\n");
gettimeofday(&tmvalEnd, NULL);
printf("%d primes, time use %d milliseconds.\n", count, ((tmvalEnd.tv_sec - tmvalBeg.tv_sec)*1000 + (tmvalEnd.tv_usec - tmvalBeg.tv_usec)/1000) );
return 0;
}
2. 筛选：
#include
#include
#include
#include
#include
#include
#define MAX_NUM 100000
static void prime(char num[], int total)
{
int i, j;
memset(num, 1, total);
for ( i = 2; i <= total/2; ++i )
{
if ( num[i] )
{
for ( j = i<<1; j <= total; j+=i )
num[j] = 0;
}
}
}
int main(int argc, char *argv[])
{
int count = 0;
struct timeval tmvalBeg, tmvalEnd;
gettimeofday(&tmvalBeg, NULL);
printf("(1, %d) prime is ", MAX_NUM);
char num[MAX_NUM];
prime(num, MAX_NUM);
int idx;
for ( idx = 2; idx < MAX_NUM; ++idx )
{
if ( num[idx] ) { ++count; printf("%u,", idx); }
}
printf("\n");
gettimeofday(&tmvalEnd, NULL);
printf("%d primes, time use %d milliseconds.\n", count, ((tmvalEnd.tv_sec - tmvalBeg.tv_sec)*1000 + (tmvalEnd.tv_usec - tmvalBeg.tv_usec)/1000) );
return 0;
} 3. 静态素数表:
#include
#include
#include
#include
#include
#define MAX_NUM 2000
static unsigned int primeTable[] = { 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,1297,1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,1429,1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,1499,1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607,1609,1613,1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709,1721,1723,1733,1741,1747,1753,1759,1777,1783,1787,1789,1801,1811,1823,1831,1847,1861,1867,1871,1873,1877,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,1979,1987,1993,1997,1999,2003 };
int main(int argc, char *argv[])
{
int count = 0;
struct timeval tmvalBeg, tmvalEnd;
gettimeofday(&tmvalBeg, NULL);
printf("(1, %d) prime is ", MAX_NUM);
unsigned int idx;
for ( idx = 0; idx < MAX_NUM; ++idx )
{
if ( primeTable[idx] < MAX_NUM )
{
++count;
printf("%d,", primeTable[idx]);
}
else break;
}
printf("\n");
gettimeofday(&tmvalEnd, NULL);
printf("%d primes, time use %d milliseconds.\n", count, ((tmvalEnd.tv_sec - tmvalBeg.tv_sec)*1000 + (tmvalEnd.tv_usec - tmvalBeg.tv_usec)/1000) );
return 0;
}
4. 6n+1与6n+5法
#include
#include
#include
#include
#include
#define MAX_NUM 100000
static unsigned int primeTable[] = { 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,1297,1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,1429,1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,1499,1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607,1609,1613,1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709,1721,1723,1733,1741,1747,1753,1759,1777,1783,1787,1789,1801,1811,1823,1831,1847,1861,1867,1871,1873,1877,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,1979,1987,1993,1997,1999,2003 };
static bool prime(unsigned int num)
{
unsigned int limit = (unsigned int)sqrt(num);
int idx = 2;
while ( primeTable[idx] <= limit )
{
if ( 0 == num % primeTable[idx] ) return false;
++idx;
}
return true;
}
int main(int argc, char *argv[])
{
int count = 2;
struct timeval tmvalBeg, tmvalEnd;
gettimeofday(&tmvalBeg, NULL);
printf("(1, %d) prime is 2,3,", MAX_NUM);
unsigned int idx;
for ( idx = 5; idx < MAX_NUM; ++idx )
{
if ( ((5 == idx % 6) || (1 == idx % 6)) && prime(idx) )
{
++count;
printf("%d,", idx);
}
}
printf("\n");
gettimeofday(&tmvalEnd, NULL);
printf("%d primes, time use %d milliseconds.\n", count, ((tmvalEnd.tv_sec - tmvalBeg.tv_sec)*1000 + (tmvalEnd.tv_usec - tmvalBeg.tv_usec)/1000) );
return 0;
}