## Prime numbers – Sieve of Eratosthenes

A prime number is a natural number with only two divisors: 1 and itself.
In this post we’re going to show how to find prime numbers using Sieve of Eratosthenes and explain how it works.

## How it works

1. Create a boolean array for the numbers you want to verify.
We create an array with size greater by one to comfortably use index as underlying number.
2. Initially mark all numbers in the array as primes.
3. Start sieving from the smallest prime number – 2.
4. Mark as non primes all multiplies of the current prime.
Notice that we start the sieve from i*i, because all numbers below have been already sieved. For example, when removing multiplies of 5 we can start from 25, because all lower multiplies of 5 have been removed when 2 and 3 were processed – 5*2 = 2*5 and 5*3 = 3*5.
5. Repeat with the next prime.
We don’t have to look for primes above maxPrime, because of reasoning in point 4.
```void markComplexNumbers(boolean[] primes) {
primes[0] = primes[1] = false;
int maxPrime = (int) Math.sqrt(primes.length);
for (int i = 2; i <= maxPrime; ++i) {
if (primes[i]) {
// remove all multiplies of this prime:
for (int j = i*i; j < primes.length; j += i) {
primes[j] = false;
}
}
}
}
```

## Properties of the Sieve of Eratosthenes

• time complexity is O(n log log n)
• memory complexity is O(n+1)
• works great for small (in mathematical sense ;-) ) prime numbers

## Complete example

```package com.farenda.tutorials.algorithms.primes;

import java.util.Arrays;

public class Sieve {

public static void main(String[] args) {
boolean[] primes = findPrimes(101);
printPrimes(primes);
}

public static boolean[] findPrimes(int to) {
// +1 to have index and number with the same value:
boolean[] primes = new boolean[to+1];
Arrays.fill(primes, true);
markComplexNumbers(primes);
return primes;
}

private static void markComplexNumbers(boolean[] primes) {
primes[0] = primes[1] = false;
int maxPrime = (int) Math.sqrt(primes.length);
for (int i = 2; i <= maxPrime; ++i) {
if (primes[i]) {
// remove all multiplies of this prime:
for (int j = i*i; j < primes.length; j += i) {
primes[j] = false;
}
}
}
}

private static void printPrimes(boolean[] primes) {
for (int i = 2; i < primes.length; ++i) {
if (primes[i]) {
System.out.print(i + ", ");
}
}
}
}
```

The above code produces the following output:

```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,
```
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