# The Sieve of Eratosthenes

A prime number is a number which only divisible by 1 and itself. Another way to think about primes is that if you have a prime number of square tiles, then they cannot be arranged to form a rectangle which is more than one tile wide. Below is an example of how 7 square tiles (a prime number) can be arranged: The "Sieve of Eratosthenes" is a method that was used in the 3rd century BC by a Greek librarian called Eratosthenes to determine which numbers are prime. Basically you write down the first N integers and cross out all multiples of 2 (i.e. 4, 6, 8, 10, etc), and then all multiples of the next number which has not yet been crossed out (i.e. 3) and so on. Any numbers which are left at the end of this procedure are primes.

The C++ program below implements this method:

 ```// // EratosthenesSieve.cpp // // Anthony Dunk, 31/3/07 // // This program uses the "Sieve of Eratosthenes" method to build a list of the prime // numbers between 1 and N. // #include "stdafx.h" int main(int argc, char* argv[]) { // // Find all primes between 1 and N // const int N = 1000; // Define the sieve bool prime[N+1]; // Flag all numbers as prime initially int i,j,p; for (i=1; i<=N; i++) prime[i]=true; // Start with first prime, 2 p = 2; while (true) { // Eliminate all factors of this prime j=2*p; while (j<=N) { prime[j]=false; // Flag this number as non-prime j=j+p; } // Find the next prime bool next_prime_found = false; for (i=p+1; i<=N; i++) { if (prime[i]) { p = i; next_prime_found = true; break; } } // Exit loop if all prime factors have been eliminated if (!next_prime_found) break; } // Display the primes and count them printf("Prime numbers between 1 and %d:\n\n",N); int count = 0; for (i=2; i<=N; i++) { if (prime[i]) { printf("%4d ",i); count++; } } printf("\n\n"); printf("%d primes\n\n",count); return 0; } ```

And here are the results when the program is run with N = 1000:

 ```Prime numbers between 1 and 1000: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 168 primes ```

If you interested in the primes I can recommend Marcus du Sautoy's The Music of the Primes as an interesting book to read.

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