# Prime Numbers

A prime number is a whole 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 represented in form of 6n+1 or 6n-1 except 2 and 3, where n is natural number.
• Two and Three are only two consecutive natural numbers which are prime too.
• 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
```
```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 n-1 and for every number check if it divides n. If we find any number that divides, we return false.

## C++

 `// A school method based C++ program to` `// check if a number is prime` `#include ` `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 n-1` `    ``for` `(``int` `i = 2; i < 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`     `def` `isPrime(n):`   `    ``# Corner case` `    ``if` `(n <``=` `1``):` `        ``return` `False`   `    ``# Check from 2 to n-1` `    ``for` `i ``in` `range``(``2``, n):` `        ``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 n-1` `        ``for` `(``int` `i = 2; i < 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

 ``

Output

``` true
```

Time complexity : O(n)

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 mumber` `// is prime or not using recursion` `#include ` `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``;` `    ``}`   `    ``// base cases` `    ``if` `(n % i == 0) ` `    ``{` `        ``return` `false``;` `    ``}` `    ``else` `    ``{` `        ``return` `true``;` `    ``}` `    ``i++;` `    ``return` `isPrime(n);` `}`   `// Driver Code` `int` `main()` `{`   `    ``isPrime(11) ? cout << ``" true\n"` `: cout << ``" false\n"``;` `    ``return` `0;` `}`   `// This code is contributed by yashbeersingh42`

Output

``` true
```

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

Algorithms to find all prime numbers smaller than the N.

More problems related to Prime number

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