- Problem Statement
- Simple Approach
- Efficient Approach: Sieve of Eratosthenes
- Dry run of the above approach
- C++ Implementation Efficient Approach
- Java Implementation of Efficient Approach
- Python Implementation of Efficient Approach
- Analysis of Time Complexity
- Conclusion
- Practice Questions
- Frequently Asked Questions

## Problem Statement

Given a number n, print all primes smaller than or equal to n. It is also given that n is a small number.

A prime number is a number that is divisible by only two numbers – themselves and 1

**Example:**

**Input:** n =10**Output:** 2 3 5 7**Explanation:** Only 4 prime numbers are there which are less than or equal to 10.

**Input: **n = 20

**Output**: 2 3 5 7 11 13 17 19

**Explanation:**All above numbers are prime which are less than or equal to 20.

## Simple Approach

Iterate from 2 to N, and check for prime. If it is a prime number, print the number. Below is the implementation of the above approach

### C++ Implementation

int checkPrime(int n) { if (n <= 1) return 0; for (int i = 2; i < n; i++) if (n % i == 0) return 0; return 1; } // Function to print all the prime numbers from // range 2 to n void printPrime(int n) { for (int i = 2; i <= n; i++) { if (checkPrime(i)) cout << i << " "; } }

### Java Implementation

class Interviewbit { static int isPrime(int n) { for (int i = 2; i < n; i++) if (n % i == 0) return 0; return 1; } static void printPrime(int n) { for (int i = 2; i <= n; i++) { if (isPrime(i)) System.out.print(i + " "); } } }

### Python Implementation

def isPrime(n): # Corner case if n <= 1 : return False # check from 2 to n-1 for i in range(2, n): if n % i == 0: return False return True # Function to print primes def printPrime(n): for i in range(2, n + 1): if isPrime(i): print(i, end = " ")

**Time complexity:** O(N*N), Where N is the number.**Space complexity:** O(1)

## Efficient Approach: Sieve of Eratosthenes

The **sieve of Eratosthenes** is one of the most efficient ways to find all primes smaller than n when n is smaller than 10 million or so. Following is the algorithm to find all the prime numbers less than or equal to a given integer *n* by the Eratosthenes method:

When the algorithm terminates, all the numbers in the list that are not marked are prime.

- Firstly write all the numbers from 2,3,4…. n
- Now take the first prime number and mark all its multiples as visited.
- Now when you move forward take another number which is unvisited yet and then follow the same step-2 with that number.
- All numbers in the list left unmarked when the algorithm ends are referred to as prime numbers.

### Dry run of the above approach

**Step 1: The numbers between 1 and 100 are listed in the table below.**

1 | 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 |

**Step 2: The next step is to write in bold all the multiples of 2, except 2 itself.**

1 | 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 |

**Step 3: Now bold all multiples of 3, 5, and 7 and only leave these numbers.**

1 | 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 |

**Step 4: Since the multiples of 11, 13, 17, and 19 are not present on the list, 1 is finally shaded because it is not prime.**

1 | 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 |

**Step 5: The unshaded numbers are prime. They include:**

2, 3, 5,7, 11, 13,17,19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

### C++ Implementation Efficient Approach

void SieveOfEratosthenes(int n) { bool prime[n + 1]; memset(prime, true, sizeof(prime)); for (int p = 2; p * p <= n; p++) { if (prime[p] == true) { for (int i = p * p; i <= n; i += p) prime[i] = false; } } for (int p = 2; p <= n; p++) if (prime[p]) cout << p << " "; }

### Java Implementation of Efficient Approach

class SieveOfEratosthenes { void sieveOfEratosthenes(int n) { boolean prime[] = new boolean[n + 1]; for (int i = 0; i <= n; i++) prime[i] = true; for (int p = 2; p * p <= n; p++){ if (prime[p] == true) { for (int i = p * p; i <= n; i += p) prime[i] = false; } } for (int i = 2; i <= n; i++) { if (prime[i] == true) System.out.print(i + " "); } } }

### Python Implementation of Efficient Approach

def SieveOfEratosthenes(n): prime = [True for i in range(n+1)] p = 2 while (p * p <= n): if (prime[p] == True): # Update all multiples of p for i in range(p * p, n+1, p): prime[i] = False p += 1 # Print all prime numbers for p in range(2, n+1): if prime[p]: print p,

**Time complexity of Sieve of Eratosthenes**:

**Space complexity:** O(1)

### Analysis of Time Complexity

**PREREQUISITE:**Time complexity of the Harmonic Series is O(logN), when N is tending to infinity.- If we analyze our sieve for larger numbers then we can see .
- The inner loop at each iteration can be expressed by:
- i=2, the inner loop will be executed → upper bound is (n/2) times
- i=3, the inner loop will be executed → upper bound is (n/3) times
- i=5, the inner loop will be executed → upper bound is (n/5) times
- …
- i=n(if prime number), the inner loop will be executed → upper bound is 1 times.

- The above series time complexity is less than harmonic series:
- n/2 + n/4 + ..+ 1 → n (1/2 + 1/3 + 1/4 + … 1/n),

- Hence, the run loop times should be
- n (1/2 + 1/3 + 1/5 + 1/7 + 1/ 11 + 1/13 + …).= logn That is why, in some article,

- The time complexity is defined in O(nloglogn).

## Conclusion

The outer loop is a general for-loop from 1 to N, the time complexity is O(N). The total time complexity is O(NloglogN).

## Practice Questions

## Frequently Asked Questions

**Q: How fast is Sieve of Eratosthenes?****A:** It is very fast as it can store prime numbers up to the 1e7 range easily.

**Q: How do you improve the sieve of Eratosthenes?****A:** We can further improve the sieve to O(n) by writing it to an iterative sieve.

**Q: How does the Sieve of Eratosthenes work?****A:** It works on the principle of cancellation of prime factors.