Apart from Sieve of Eratosthenes method to generate Prime numbers, we can implement a new Algorithm for generating prime numbers from **1** to **N**.

It might be amazing to know that **all the prime numbers ≥ 5** can be traced from a pattern:

Let’s try to understand the series:

Series 1:

5 + 6 = 11

11 + 6 = 17

17 + 6 = 23

23 + 6 = 29

…

…

Series 2:

7 + 6 = 13

13 + 6 = 19

…

…

We see that adding **6** to **5** and again adding **6** to the obtained result, we are able to get another prime number. Similarly, adding **6** to **7** and again adding **6** to the obtained result. we can get another prime number. Hence, we can inference that we can obtain all the primes numbers doing same procedure.

However, when we keep adding **6** to the previous result to obtain next prime number, we can notice some numbers that are not prime numbers.

For example:

13 + 6 = 19 (Prime Number)

19 + 6 = 25 (Composite Number; Say Pseudo Prime Number)

29 + 6 = 35 (Composite Number; Say Pseudo Prime Number)

We may say these types of Composite Numbers as Pseudo Prime Numbers (appear as prime number, but are not in reality).

Now, if we are able to separate these Pseudo Prime Numbers from Real Prime Numbers, we can get the Real Prime Numbers. Using 4 important equations, we can exactly track all these Pseudo Prime Numbers and separate them from Real Prime Numbers.

The Equation that will generate series is:

y = 6 * x ± 1

where x = 1, 2, 3, 4, 5, 6, 7, …

For example:

6 * (1) – 1 = 5 (Prime Number)

6 * (1) + 1 = 7 (Prime Number)

6 * (2) – 1 = 11 (Prime Number)

6 * (2) + 1 = 13 (Prime Number)

6 * (3) – 1 = 17 (Prime Number)

6 * (3) + 1 = 19 (Prime Number)

6 * (4) – 1 = 23 (Prime Number)

6 * (4) + 1 = 25 (Pseudo Prime Number)

We can track all the Pseudo Prime Numbers using 4 equations:

**For the series produced from y = 6 * x – 1:**

**y = (6 * d * x) + d**where, x = {1, 2, 3, 4, 5, 6, …}**y = (6 * d * x) + (d * d)**where, x = {0, 1, 2, 3, 4, 5, …} and d = {5, (5+6), (5+6+6), (5+6+6+6), …, (5+(n-1)*6)}

**For the series produced from y = 6 * x + 1:**

**y = (6 * d * x) + (d * d)**where, x = {0, 1, 2, 3, …}**y = (6 * d * x) + (d * d) – 2 * d**where, x = {1, 2, 3, 4, …}, d = {7, (7+6), (7+6+6), …, (7 + (n – 1) * 6)} and n = {1, 2, 3, 4, 5, …}

Examples

Take d = 5 and x = 0 in equation 2,

y = 25 (Pseudo Prime Number)

Take d = 5 and x = 1 in equation 1,

y = 35 (Pseudo Prime Number)

Similarly, putting values in equation 3 and 4 we can track all Pseudo Prime Numbers.

**How to implement in programming?**

We assume two bool type arrays of same size.

Say first one isarray1, every element of which is initialized to0.

And, second one isarray2, every element of which is initialized to1.

Now,

Inarray1, we initialize the specified indexes with1, the index values are calculated using the equationy = (6 * x) ± 1.

And, inarray2, we initialize the specified indexes with0, the index values are calculated using 4 equations described above.Now, run a loop from

0toNand print the index value of array,

Ifarray1[i] = 1andarray2[i] = 1(i is a prime number).

Below is the implementation of the approach:

`// C++ implementation of the approach ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to return the count of prime numbers <= n ` `int` `countPrimesUpto(` `int` `n) ` `{ ` ` ` ` ` `// To store the count of prime numbers ` ` ` `int` `count = 0; ` ` ` ` ` `// To mark all the prime numbers according ` ` ` `// to the equation (y = 6*x -/+ 1) ` ` ` `// where x = 1, 2, 3, 4, 5, 6, 7, ... ` ` ` `bool` `arr1[n + 1]; ` ` ` ` ` `// To mark all the Pseudo Prime Numbers ` ` ` `// using the four equations ` ` ` `// described in the approach ` ` ` `bool` `arr2[n + 1]; ` ` ` ` ` `// Starting with >= 5 ` ` ` `int` `d = 5; ` ` ` ` ` `// 2 and 3 are primes ` ` ` `arr1[2] = arr2[2] = 1; ` ` ` `arr1[3] = arr2[3] = 1; ` ` ` ` ` `// Initialize every element of arr1 with 0 ` ` ` `memset` `(arr1, 0, ` `sizeof` `(arr1)); ` ` ` ` ` `// Initialize every element of arr2 with 1 ` ` ` `memset` `(arr2, 1, ` `sizeof` `(arr2)); ` ` ` ` ` `// Update arr1[] to mark all the primes ` ` ` `while` `(d <= n) { ` ` ` ` ` `// For 5, (5 + 6), (5 + 6 + 6), ... ` ` ` `memset` `(arr1 + d, 1, (` `sizeof` `(arr1)) / (n + 1)); ` ` ` ` ` `// For 7, (7 + 6), (7 + 6 + 6), ... ` ` ` `memset` `(arr1 + (d + 2), 1, (` `sizeof` `(arr1)) / (n + 1)); ` ` ` ` ` `// Increase d by 6 ` ` ` `d = d + 6; ` ` ` `} ` ` ` ` ` `// Update arr2[] to mark all pseudo primes ` ` ` `for` `(` `int` `i = 5; i * i <= n; i = i + 6) { ` ` ` `int` `j = 0; ` ` ` ` ` `// We will run while loop until we find all ` ` ` `// pseudo prime numbers <= n ` ` ` `while` `(1) { ` ` ` `int` `flag = 0; ` ` ` ` ` `// Equation 1 ` ` ` `int` `temp1 = 6 * i * (j + 1) + i; ` ` ` ` ` `// Equation 2 ` ` ` `int` `temp2 = ((6 * i * j) + i * i); ` ` ` ` ` `// Equation 3 ` ` ` `int` `temp3 = ((6 * (i + 2) * j) ` ` ` `+ ((i + 2) * (i + 2))); ` ` ` ` ` `// Equation 4 ` ` ` `int` `temp4 = ((6 * (i + 2) * (j + 1)) ` ` ` `+ ((i + 2) * (i + 2)) - 2 * (i + 2)); ` ` ` ` ` `// If obtained pseudo prime number <=n then its ` ` ` `// corresponding index in arr2 is set to 0 ` ` ` ` ` `// Result of equation 1 ` ` ` `if` `(temp1 <= n) { ` ` ` `arr2[temp1] = 0; ` ` ` `} ` ` ` `else` `{ ` ` ` `flag++; ` ` ` `} ` ` ` ` ` `// Result of equation 2 ` ` ` `if` `(temp2 <= n) { ` ` ` `arr2[temp2] = 0; ` ` ` `} ` ` ` `else` `{ ` ` ` `flag++; ` ` ` `} ` ` ` ` ` `// Result of equation 3 ` ` ` `if` `(temp3 <= n) { ` ` ` `arr2[temp3] = 0; ` ` ` `} ` ` ` `else` `{ ` ` ` `flag++; ` ` ` `} ` ` ` ` ` `// Result of equation 4 ` ` ` `if` `(temp4 <= n) { ` ` ` `arr2[temp4] = 0; ` ` ` `} ` ` ` `else` `{ ` ` ` `flag++; ` ` ` `} ` ` ` ` ` `if` `(flag == 4) { ` ` ` `break` `; ` ` ` `} ` ` ` `j++; ` ` ` `} ` ` ` `} ` ` ` ` ` `// Include 2 ` ` ` `if` `(n >= 2) ` ` ` `count++; ` ` ` ` ` `// Include 3 ` ` ` `if` `(n >= 3) ` ` ` `count++; ` ` ` ` ` `// If arr1[i] = 1 && arr2[i] = 1 then i is prime number ` ` ` `// i.e. it is a prime which is not a pseudo prime ` ` ` `for` `(` `int` `p = 5; p <= n; p = p + 6) { ` ` ` `if` `(arr2[p] == 1 && arr1[p] == 1) ` ` ` `count++; ` ` ` ` ` `if` `(arr2[p + 2] == 1 && arr1[p + 2] == 1) ` ` ` `count++; ` ` ` `} ` ` ` ` ` `return` `count; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `n = 100; ` ` ` `cout << countPrimesUpto(n); ` `} ` |

*chevron_right*

*filter_none*

**Output:**

25

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the **DSA Self Paced Course** at a student-friendly price and become industry ready.

## Recommended Posts:

- How to generate Large Prime numbers for RSA Algorithm
- Print numbers such that no two consecutive numbers are co-prime and every three consecutive numbers are co-prime
- Algorithm to generate positive rational numbers
- Absolute Difference between the Sum of Non-Prime numbers and Prime numbers of an Array
- Absolute difference between the Product of Non-Prime numbers and Prime numbers of an Array
- Absolute difference between the XOR of Non-Prime numbers and Prime numbers of an Array
- Count prime numbers that can be expressed as sum of consecutive prime numbers
- Sum of two numbers if the original ratio and new ratio obtained by adding a given number to each number is given
- Generate K co-prime pairs of factors of a given number
- Print the nearest prime number formed by adding prime numbers to N
- Check if a prime number can be expressed as sum of two Prime Numbers
- Check if a number is Prime, Semi-Prime or Composite for very large numbers
- Count numbers in a given range whose count of prime factors is a Prime Number
- Program to find the Nth Prime Number
- Prime numbers after prime P with sum S
- Print prime numbers with prime sum of digits in an array
- Print all numbers whose set of prime factors is a subset of the set of the prime factors of X
- Sum of prime numbers without odd prime digits
- Permutation of first N positive integers such that prime numbers are at prime indices
- Permutation of first N positive integers such that prime numbers are at prime indices | Set 2

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.