Postulado de Bertrand

En matemáticas, el Postulado de Bertrand establece que hay un número primo en el rango  donde  n es un número natural y n >= 4. Chebyshev y luego Ramanujan lo demostraron. Una forma indulgente del postulado establece que existe un número primo en el rango de n a 2n para cualquier n(n >= 2).
 

Existe un primo p para  para todo n <= 4. La forma menos estricta establece que existe un primo p. Para  todo n <= 2. 
Ejemplos: 
Para n = 4 y 2*n – 2 = 6, 
5 es un número primo en el rango (4, 6).
Para n = 5 y 2*n – 2 = 8, 
7 es un número primo en el rango (5, 8).
Para n = 6 y 2*n – 2 = 10, 
7 es un número primo en el rango (6, 10).
Para n = 7 y 2*n – 2 = 12, 
11 es un número primo en el rango (7, 12).
Para n = 8 y 2*n – 2 = 14, 
11 es un número primo en el rango (8, 14). 
 

Ejemplos: 
 

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

// 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;
}

// 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

# 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# 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
// 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
?>

<script>
 
    // Javascript code to verify Bertrand's
    // postulate for a given n.
     
    function isprime(n)
    {
        // check whether a number
        // is prime or not
        for (let i = 2; i * i <= n; i++)
            if (n % i == 0) // i is a factor of n
                return false;
        return true;
    }
     
    let n = 10;
   
    // Checking Bertrand's postulate
    // Presence of prime numbers in
    // range (n, 2n - 2)
    document.write(
    "Prime numbers in range (" + n + ", "+ (2 * n - 2) + ")" +
    "</br>"
    );
    for (let i = n + 1; i < 2 * n - 2; i++)
        if (isprime(i))
            document.write(i + "</br>");
 
         
</script>

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