- Sieve of Eratosthenes
- In Mathematics, the
sieve of Eratosthenes,
is a simple, ancient algorithm
for finding all prime numbers
up to a specified integer N
- The algorithm is named after
Eratosthenes of Cyrene,
an ancient Greek mathematician.
The sieve algorithm was described and attributed to Eratosthenes in the Introduction to Arithmetic by Nicomachus.
- Reference: Wikipedia
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- Illustrative example:
find all primes upto 120
- Start with the following
candidates:
2 3 4 5 6 7 8 9 ..... 120
- Let's re-arrange it in a way that we
can display all candidates:
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
- Find the
first unmarked number:
2
Delete all number that are divisible by 2:
2 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119
- Find the
next unmarked number:
3
Delete all number that are divisible by 3:
2 3 5 7 11 13 17 19 23 25 29 31 35 37 41 43 47 49 53 55 59 61 65 67 71 73 77 79 83 85 89 91 95 97 101 103 107 109 113 115 119
- Find the
next unmarked number:
5
Delete all number that are divisible by 5:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 49 53 59 61 67 71 73 77 79 83 89 91 97 101 103 107 109 113 119
- And so on...
Repeat until you process all numbers in the list.
The red numbers (marked) are prime numbers
- Start with the following
candidates:
- Here is a nice animation from
Wikipedia:
- Problem description:
- Write a method
sieveOfE() with:
- input:
the number N
- output:
a boolean array
The array element with index i is true if the number i is prime and false otherwise.
- input:
the number N
- Write a method
sieveOfE() with:
- Method header:
public class PrimeSieve { public static boolean[] sieveOfE( int N ) { .... } }
- Pseudo code:
boolean[] out = new boolean[ N + 1 ]; /* ------------------------ Initialize out[] ------------------------ */ out[0] = false; out[1] = false; set all other values out[i] = true; // They are all candidates /* ------------------------ Perform the sieve of E ------------------------ */ Find the next value out[i] == true; Set: out[2*i] = false, out[3*i] = false, .... (upto out[N]) Find the next value out[i] == true; Set: out[2*i] = false, out[3*i] = false, .... (upto out[N]) And so on...
- Refined pseudo code:
boolean[] out = new boolean[ N + 1 ]; /* ------------------------ Initialize out[] ------------------------ */ out[0] = false; out[1] = false; for ( i = 2; i <= N; i++ ) out[i] = true; // They are all candidates /* ------------------------ Perform the sieve of E ------------------------ */ k = 1; // Start before the first prime number while ( k <= N ) { Starting from k+1, find next prime number i; Set: out[2*i] = false, out[3*i] = false, .... (upto out[N]) k = i; }
- Java Method:
public static boolean[] sieveOfE ( int N ) { boolean[] isPrime = new boolean[ N + 1 ]; // Return variable int i, j, k; /* ------------------------ Initialize isPrime[] ------------------------ */ isPrime[0] = false; // 0 is not prime isPrime[1] = false; // 1 is not prime for ( i = 2; i <= N; i++ ) isPrime[i] = true; // All other number are possible candidates /* ------------------------ Perform the sieve of E ------------------------ */ k = 2; // Start with 2 to find all primes while ( k <= N ) { /* ---------------------------------------- Starting from k, find next prime number number i A prime number is detected by: isPrime[i] == true ---------------------------------------- */ for ( i = k; i <= N; i++ ) if ( isPrime[i] ) break; // Found ! /* -------------------------------------- Set: isPrime[2*i] = false, isPrime[3*i] = false, .... (upto isPrime[N]) -------------------------------------- */ for ( j = 2*i; j <= N; j = j + i ) isPrime[j] = false; k = i+1; // Set up k for next iteration !!! } return isPrime; }
- Example Program:
(Demo above code)
- Prog file: click here
How to run the program:
- Right click on link and
save in a scratch directory
- To compile: javac Primes1.java
- To run: java Primes1