Prime Numbers

A prime number is a natural number greater than 1, which is only divisible by 1 and itself. First few prime numbers are : 2 3 5 7 11 13 17 19 23 …..

Some interesting fact about Prime numbers 

• Two is the only even Prime number.
• Every prime number can be represented in form of 6n+1 or 6n-1 except the prime number 2 and 3, where n is a natural number.
• Two and Three are only two consecutive natural numbers that are prime.
• Goldbach Conjecture: Every even integer greater than 2 can be expressed as the sum of two primes.
• Wilson Theorem: Wilson’s theorem states that a natural number p > 1 is a prime number if and only if

(p - 1) ! ≡  -1   mod p 
OR  (p - 1) ! ≡  (p-1) mod p

• Fermat’s Little Theorem: If n is a prime number, then for every a, 1 <= a < n,

an-1 ≡ 1 (mod n)
OR 
an-1 % n = 1

• Prime Number Theorem: The probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n.
• Lemoine’s Conjecture: Any odd integer greater than 5 can be expressed as a sum of an odd prime (all primes other than 2 are odd) and an even semiprime. A semiprime number is a product of two prime numbers. This is called Lemoine’s conjecture.

How we check whether a number is Prime or not?

• Naive solution. 
  A naive solution is to iterate through all numbers from 2 to sqrt(n) and for every number check if it divides n. If we find any number that divides, we return false.

C++14

// A school method based C++ program to
// check if a number is prime
#include <bits/stdc++.h>
using namespace std;

// function check whether a number
// is prime or not
bool isPrime(int n)
{
    // Corner case
    if (n <= 1)
        return false;

    // Check from 2 to square root of n
    for (int i = 2; i <= sqrt(n); i++)
        if (n % i == 0)
            return false;

    return true;
}

// Driver Code
int main()
{
    isPrime(11) ? cout << " true\n" : cout << " false\n";
    return 0;
}

Java

// A school method based Java program to
// check if a number is prime
import java.util.*;
import java.lang.*;

class GFG {

    // Check for number prime or not
    static boolean isPrime(int n)
    {

        // Check if number is less than
        // equal to 1
        if (n <= 1)
            return false;

        // Check if number is 2
        else if (n == 2)
            return true;

        // Check if n is a multiple of 2
        else if (n % 2 == 0)
            return false;

        // If not, then just check the odds
        for (int i = 3; i <= Math.sqrt(n); i += 2)
        {
            if (n % i == 0)
                return false;
        }
        return true;
    }

    // Driver code
    public static void main(String[] args)
    {
        if (isPrime(19))
            System.out.println("true");

        else
            System.out.println("false");
    }
}

// This code is contributed by Ronak Bhensdadia

Python3

# A school method based Python3 program
# to check if a number is prime

# function check whether a number
# is prime or not

#import sqrt from math module
from math import sqrt

def isPrime(n):

    # Corner case
    if (n <= 1):
        return False

    # Check from 2 to sqrt(n)
    for i in range(2, int(sqrt(n))+1):
        if (n % i == 0):
            return False

    return True


# Driver Code
if isPrime(11):
    print("true")
else:
    print("false")

# This code is contributed by Sachin Bisht

C#

// A school method based C# program to
// check if a number is prime
using System;

class GFG {
    // function check whether a
    // number is prime or not
    static bool isPrime(int n)
    {
        // Corner case
        if (n <= 1)
            return false;

        // Check from 2 to sqrt(n)
        for (int i = 2; i < Math.Sqrt(n); i++)
            if (n % i == 0)
                return false;

        return true;
    }

    // Driver Code
    static void Main()
    {
        if (isPrime(11))
            Console.Write(" true");

        else
            Console.Write(" false");
    }
}

// This code is contributed by Sam007

PHP

<?php
// A school method based PHP program to
// check if a number is prime

// function check whether a number
// is prime or not
function isPrime($n)
{
    // Corner case
    if ($n <= 1)
        return false;

    // Check from 2 to n-1
    for ($i = 2; $i < $n; $i++)
        if ($n % $i == 0)
            return false;

    return true;
}

// Driver Code
if(isPrime(11))
    echo("true");
else
    echo("false");

// This code is contributed by Ajit.
?>

Javascript

<script>

// A school method based Javascript program to
// check if a number is prime


// function check whether a number
// is prime or not
function isPrime(n)
{
    // Corner case
    if (n <= 1)
        return false;

    // Check from 2 to n-1
    for (let i = 2; i < n; i++)
        if (n % i == 0)
            return false;

    return true;
}

// Driver Code

    isPrime(11) ? document.write(" true" + "<br>") : document.write(" false" + "<br>");

// This code is contributed by Mayank Tyagi

</script>

Output

 true

Time Complexity: O() 

Using Recursion :

Recursion can also be used to check if a number between 2 to n – 1 divides n. If we find any number that divides, we return false.

C++

// C++ program to check whether a number
// is prime or not using recursion
#include <iostream>
using namespace std;

// function check whether a number
// is prime or not
bool isPrime(int n)
{
    static int i = 2;

    // corner cases
    if (n == 0 || n == 1) {
        return false;
    }

    // Checking Prime
    if (n == i)
        return true;

    // base cases
    if (n % i == 0) {
        return false;
    }
    i++;
    return isPrime(n);
}

// Driver Code
int main()
{

    isPrime(35) ? cout << " true\n" : cout << " false\n";
    return 0;
}

// This code is contributed by yashbeersingh42

Java

// Java program to check whether a number
// is prime or not using recursion
class GFG{

static int i = 2;

// Function check whether a number
// is prime or not
public static boolean isPrime(int n)
{

    // Corner cases
    if (n == 0 || n == 1)
    {
        return false;
    }

    // Checking Prime
    if (n == i)
        return true;

    // Base cases
    if (n % i == 0)
    {
        return false;
    }
    i++;
    return isPrime(n);
}

// Driver Code
public static void main(String[] args)
{
    if (isPrime(35))
    {
        System.out.println("true");
    }
    else
    {
        System.out.println("false");
    }
}
}

// This code is contributed by divyeshrabadiya07

Python3

# Python3 program to check whether a number
# is prime or not using recursion

# Function check whether a number
# is prime or not
def isPrime(n, i):

    # Corner cases
    if (n == 0 or n == 1):
        return False

    # Checking Prime
    if (n == i):
        return True

    # Base cases
    if (n % i == 0):
        return False

    i += 1

    return isPrime(n, i)

# Driver Code
if (isPrime(35, 2)):
  print("true")
else:
  print("false")

#  This code is contributed by bunnyram19

C#

// C# program to check whether a number
// is prime or not using recursion
using System;
class GFG {

    static int i = 2;

    // function check whether a number
    // is prime or not
    static bool isPrime(int n)
    {

        // corner cases
        if (n == 0 || n == 1) {
            return false;
        }

        // Checking Prime
        if (n == i)
            return true;

        // base cases
        if (n % i == 0) {
            return false;
        }
        i++;
        return isPrime(n);
    }

  static void Main() {
    if(isPrime(35))
    {
        Console.WriteLine("true");
    }
    else{
        Console.WriteLine("false");
    }
  }
}

// This code is contributed by divyesh072019

Javascript

<script>
      // JavaScript program to check whether a number
      // is prime or not using recursion

      // function check whether a number
      // is prime or not
      function isPrime(n) {
        var i = 1;

        // corner cases
        if (n == 0 || n == 1) {
          return false;
        }

        // Checking Prime
        if (n == i) return true;

        // base cases
        if (n % i == 0) {
          return false;
        }
        i++;
        return isPrime(n);
      }

      // Driver Code

      isPrime(35) ? document.write(" true\n") : document.write(" false\n");

      // This code is contributed by rdtank.
    </script>

Output

 false

Time Complexity :O(N), Space Complexity :O(N) 

• Efficient solutions
  • Primality Test | Set 1 (Introduction and School Method)
  • Primality Test | Set 2 (Fermat Method)
  • Primality Test | Set 3 (Miller–Rabin)
  • Primality Test | Set 4 (Solovay-Strassen)
  • Lucas Primality Test

Algorithms to find all prime numbers smaller than the N. 

• Sieve of Eratosthenes
• Sieve of Eratosthenes in 0(n) time complexity
• Segmented Sieve
• Sieve of Sundaram
• Bitwise Sieve
• Recent Articles on Sieve!

More problems related to Prime number 

• Find two distinct prime numbers with a given product
• Print all prime numbers less than or equal to N
• Recursive program for prime number
• Find two prime numbers with a given sum
• Find the highest occurring digit in prime numbers in a range
• Prime Factorization using Sieve O(log n) for multiple queries
• Program to print all prime factors of a given number
• Least prime factor of numbers till n
• Prime factors of LCM of array elements – GeeksforGeeks
• Program for Goldbach’s Conjecture
• Prime numbers and Fibonacci
• Composite Number
• Recent Articles on Prime Numbers!