# 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 9 11 13 17 19 23 …..

Some interesting fact about Prime numbers

1. 2 is the only even Prime number.
2. every prime number can represented in form of 6n+1 or 6n-1, where n is natural number.
3. 2, 3 are only two consecutive natural numbers which are prime too.
4. Goldbach Conjecture: Every even integer greater than 2 can be expressed as the sum of two primes.
5. GCD of a natural number with Prime is always one.
6. 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```
7. Fermat’s Little Theorem: If n is a prime number, then for every a, 1 <= a < n,
```an-1 ≡ 1 (mod n)
OR
an-1 % n = 1 ```
8. 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.

How we check whether a number is Prime or not?

1. 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 <bits/stdc++.h>
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 Program to test above function
int main()
{
isPrime(11) ? cout << " true\n" :
cout << " false\n";
return 0;
}
```

## 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 Program to test above function
if isPrime(11):
print ("true")
else:
print ("false")

# This code is contributed by Sachin Bisht
```

Output:
Time complexity :O(n)

```True
```
2. Efficient solutions

Algorithms to find all prime number smaller the N.

More problems related to Prime number

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