Bertrand’s Postulate

In mathematics, Bertrand’s Postulate states that there is a prime number in the range n to 2n - 2 where n is a natural number and n >= 4. It has been proved by Chebyshev and later by Ramanujan. A lenient form of the postulate states that there exists a prime in range n to 2n for any n(n >= 2).

There exists a prime p for n < p < 2*n - 2 for all n <= 4. The less stricter form states that there exists a prime p. For n < p < 2*n for all n <= 2.

Examples:
For n = 4 and 2*n – 2 = 6,
5 is a prime number in the range (4, 6).



For n = 5 and 2*n – 2 = 8,
7 is a prime number in the range (5, 8).

For n = 6 and 2*n – 2 = 10,
7 is a prime number in the range (6, 10).

For n = 7 and 2*n – 2 = 12,
11 is a prime number in the range (7, 12).

For n = 8 and 2*n – 2 = 14,
11 is a prime number in the range (8, 14).


Examples :

Input: n = 4
Output: Prime numbers in range (4, 6)
        5

Input: n = 5
Output: Prime numbers in range (5, 8)
        7

Input: n = 6
Output: Prime numbers in range (6, 10)
        7

C++

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// CPP code to verify Bertrand's postulate
// for a given n.
#include <bits/stdc++.h>
using namespace std;
  
bool isprime(int n)
{
    // check whether a number is prime or not
    for (int i = 2; i * i <= n; i++)
        if (n % i == 0) // i is a factor of n
            return false;
    return true;
}
  
int main()
{
    int n = 10;
  
    // Checking Bertrand's postulate
    // Presence of prime numbers in range (n, 2n - 2)
    cout << "Prime numbers in range (" << n << ", " 
         << 2 * n - 2 << ")\n";
    for (int i = n + 1; i < 2 * n - 2; i++)
        if (isprime(i))
            cout << i << "\n";
  
    return 0;
}

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Java

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// Java code to verify Bertrand's 
// postulate for a given n.
import java.io.*;
  
class GFG 
{
static boolean isprime(int n)
{
    // check whether a number
    // is prime or not
    for (int i = 2; i * i <= n; i++)
        if (n % i == 0) // i is a factor of n
            return false;
    return true;
}
  
    // Driver Code
    public static void main (String[] args) 
    {
        int n = 10;
  
        // Checking Bertrand's postulate
        // Presence of prime numbers in
        // range (n, 2n - 2)
        System.out.println("Prime numbers in range ("
                          n + ", "+ (2 * n - 2) + ")");
        for (int i = n + 1; i < 2 * n - 2; i++)
            if (isprime(i))
                System.out.println(i);
    }
}
  
// This code is contributed 
// by shiv_bhakt

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Python3

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# PHP code to verify 
# Bertrand's postulate 
# for a given n.
def isprime(n):
      
    # check whether a number
    # is prime or not
    i = 2;
    while(i * i <= n):
        if (n % i == 0):
              
            # i is a factor of n
            return False;
        i = i + 1;
    return True;
  
# Driver Code
n = 10;
  
# Checking Bertrand's 
# postulate Presence 
# of prime numbers in
# range (n, 2n - 2)
print("Prime numbers in range (" , n , 
               ", ", 2 * n - 2 , ")");
i = n + 1;
while(i < (2 * n - 2)):
    if (isprime(i)):
        print(i);
    i = i + 1;
  
# This code is contributed by mits

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C#

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// C# code to verify Bertrand's 
// postulate for a given n.
using System;
  
class GFG
{
static bool isprime(int n)
{
    // check whether a number
    // is prime or not
    for (int i = 2; i * i <= n; i++)
        if (n % i == 0) // i is a factor of n
            return false;
    return true;
}
  
// Driver Code
public static void Main () 
{
    int n = 10;
  
    // Checking Bertrand's postulate
    // Presence of prime numbers in
    // range (n, 2n - 2)
    Console.WriteLine("Prime numbers in range ("
                     n + ", "+ (2 * n - 2) + ")");
    for (int i = n + 1; i < 2 * n - 2; i++)
        if (isprime(i))
            Console.WriteLine(i);
}
}
  
// This code is contributed 
// by shiv_bhakt

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PHP

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<?php
// PHP code to verify Bertrand's
// postulate for a given n.
function isprime($n)
{
    // check whether a number 
    // is prime or not
    for ($i = 2; $i * $i <= $n; $i++)
        if ($n % $i == 0) // i is a factor of n
            return false;
    return true;
}
  
// Driver Code
$n = 10;
  
// Checking Bertrand's postulate
// Presence of prime numbers in
// range (n, 2n - 2)
echo "Prime numbers in range (" , $n
             ", ", 2 * $n - 2 , ")\n";
for ($i = $n + 1; $i < 2 * $n - 2; $i++)
    if (isprime($i))
        echo $i , "\n";
  
// This code is contributed by ajit
?>

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Output :

Prime numbers in range (10, 18)
11
13
17


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