# 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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