Bertrand’s Postulate

In mathematics, Bertrand’s Postulate states that there is a prime number in the range to 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 for all n <= 4. The less stricter form states that there exists a prime p. For 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++

 // CPP code to verify Bertrand's postulate // for a given n. #include 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; }

Java

 // 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

Python3

 # 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

C#

 // 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

PHP



Output :

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

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