×
Create a new article
We currently have 3,189 articles on s23. Type your article name above or create one of the articles listed here!

3,189Articles
in:

# Prime Numbers

Historically, the most efficient way to find all of the small primes (say all those less than 10,000,000) is by using the Sieve of Eratosthenes (ca 240 BC):

Make a list of all the integers less than or equal to n (and greater than one). Strike out the multiples of all primes less than or equal to the square root of n, then the numbers that are left are the primes.

This method is so fast that there is no reason to store a large list of primes on a computer--an efficient implementation can find them faster than a computer can read from a disk.

---

A compact prime sieve (Eratosthenes)

52 lines of portable ANSI C. Outputs roughly first and last sqrt(N) primes when searching primes up to N (rounded up to next multiple of 30), and finally reports #primes found.

```#include <stdio.h>
#include <stdlib.h>
#define DO(P,R,I,M,E,S,i0,v0,i1,v1,i2,v2,i3,v3,i4,v4,i5,v5,i6,v6,i7,v7) k=P;\
if (!(sieve[n] & (1<<R)))\
{ printf(" %ld",30*n + bits[R]);\
e = eos - I*n - M;\
for (m = sieve + (30*n + E) * n + S; m < e; m += i0)\
{ *m        |= (1<<v0); *(m += i1) |= (1<<v1);\
*(m +=i2) |= (1<<v2); *(m += i3) |= (1<<v3);\
*(m +=i4) |= (1<<v4); *(m += i5) |= (1<<v5);\
*(m +=i6) |= (1<<v6); *(m += i7) |= (1<<v7);\
}\
if (m < eos) { *m|=(1<<v0);\
if ((m += i1) < eos) { *m |= (1<<v1);\
if ((m += i2) < eos) { *m |= (1<<v2);\
if ((m += i3) < eos) { *m |= (1<<v3);\
if ((m += i4) < eos) { *m |= (1<<v4);\
if ((m += i5) < eos) { *m |= (1<<v5);\
if ((m += i6) < eos)   *m |= (1<<v6);\
} } } } } } }
char bits[] = {1,7,11,13,17,19,23,29} ;

int main(int argc, char *argv[])
{
unsigned long p,q,r,k=0,n,s;
char *m,*e,*eos,*sieve;
long bytes,atol();

if (argc!=2) printf("usage: %s <bytes_used or -maxprime>\n",*argv), exit(0);
if ((bytes=atol(argv[1])) < 0) bytes = 1 + (-bytes)/30;
if (!(sieve = malloc(bytes))) printf("Out of memory.\n"), exit(0);
if (bytes > 30) for (k = r = (bytes-1)/30; (q = r/k) < k; k >>= 1) k += q;
eos = sieve + bytes; s = k + 1; *sieve = 1; printf("2 3 5");
for (n = p = q = r = 0; n < s; n++)
{ DO(p++,0,28, 0, 2, 0,p,0,r,1,q,2,k,3,q,4,k,5,q,6,r,7); r++;
DO(q++,1,24, 6,14, 1,r,5,q,4,p,0,k,7,p,3,q,2,r,6,p,1); r++; q++;
DO(p-1,2,26, 9,22, 4,q,0,k,6,q,1,k,7,q,3,r,5,p,2,r,4); r++;
DO(q-1,3,28,12,26, 5,p,5,q,2,p,1,k,7,r,4,p,3,r,0,k,6);
DO(q+1,4,26,15,34, 9,q,5,p,6,k,0,r,3,p,4,r,7,k,1,p,2); r++;
DO(p+1,5,28,17,38,12,k,0,q,4,r,2,p,5,r,3,q,7,k,1,q,6); r++; q++;
DO(q++,6,26,20,46,17,k,5,r,1,p,6,r,2,k,3,p,7,q,0,p,4); r++;
DO(p++,7,24,23,58,28,r,0,k,7,r,6,q,5,p,4,q,3,p,2,q,1);
}
printf(" ...");
for (p = bytes - s; p < bytes; p++)
for (k = 0; k < 8; k++)
if (!(sieve[p] & (1<<k))) printf(" %ld",30 * p + bits[k]);
for (p = 0, n=3; p < bytes; p++)
for (k = 0; k < 8; k++) n += !(sieve[p] & (1<<k));
printf("\n%ld primes found\n", n);
exit(0);
}
```
```compile with: gcc -O sieve.c -o sieve
```
```usage: ./sieve <bytes_used or -maxprime>
```
```example: ./sieve 523
```

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 ... 15541 15551 15559 15569 15581 15583 15601 15607 15619 15629 15641 15643 15647 15649 15661 15667 15671 15679 15683

1831 primes found

siehe auch (deutsch): Primzahl

Cookies help us deliver our services. By using our services, you agree to our use of cookies.

### Recent changes

• BruderSpektrum • 1 month ago
• BruderSpektrum • 1 month ago
• BruderSpektrum • 1 month ago
• BruderSpektrum • 1 month ago
• Cookies help us deliver our services. By using our services, you agree to our use of cookies.