# Check if a number is Fermat Pseudoprime

Given a number N and a base number A. The task is to check whether the number is a Fermat Pseudoprime to the base.
The number N is called as Fermat Pseudoprime to the base A, if

1. A > 1
2. N is a composite number
3. N divides AN-1 – 1.

Examples:

Input : N = 645, a = 2
Output :1
645 = 3*5*43, Hence it is a composite number
Also 645 divides 2^(644)-1
Hence it is a Fermat Pseudoprime.

Input : N = 6, a = 2
Output :0

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: The approach is to check the below conditions:

If all of the above conditions satisfy then N is a fermat pseudoprime to base A.

Below is the implementation of the above approach:

## C++

 `// C++ program to check if N is Fermat pseudoprime ` `// to the base A or not ` `#include ` `using` `namespace` `std; ` ` `  `// Function to check if the given number is composite ` `bool` `checkcomposite(``int` `n) ` `{ ` `    ``// Check if there is any divisor of n less than sqrt(n) ` `    ``for` `(``int` `i = 2; i <= ``sqrt``(n); i++) { ` `        ``if` `(n % i == 0) ` `            ``return` `1; ` `    ``} ` `    ``return` `0; ` `} ` ` `  `// Effectively calculate (x^y) modulo mod ` `int` `power(``int` `x, ``int` `y, ``int` `mod) ` `{ ` ` `  `    ``// Initialize result ` `    ``int` `res = 1; ` ` `  `    ``while` `(y) { ` ` `  `        ``// If power is odd, then update the answer ` `        ``if` `(y & 1) ` `            ``res = (res * x) % mod; ` ` `  `        ``// Square the number and reduce ` `        ``// the power to its half ` `        ``y = y >> 1; ` `        ``x = (x * x) % mod; ` `    ``} ` ` `  `    ``// Return the result ` `    ``return` `res; ` `} ` ` `  `// Function to check for Fermat Pseudoprime ` `bool` `Check(``int` `n, ``int` `a) ` `{ ` ` `  `    ``// If it is composite and satisfy Fermat criterion ` `    ``if` `(a>1 && checkcomposite(n) && power(a, n - 1, n) == 1) ` `        ``return` `1; ` ` `  `    ``// Else return 0 ` `    ``return` `0; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` ` `  `    ``int` `N = 645; ` `    ``int` `a = 2; ` `     `  `   ``//  Function call ` `    ``cout << Check(N, a); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to check if N is Fermat pseudoprime  ` `// to the base A or not  ` `class` `GFG  ` `{ ` ` `  `    ``// Function to check if  ` `    ``// the given number is composite  ` `    ``static` `boolean` `checkcomposite(``int` `n)  ` `    ``{ ` `        ``// Check if there is any divisor of n  ` `        ``// less than sqrt(n)  ` `        ``for` `(``int` `i = ``2``; i <= Math.sqrt(n); i++)  ` `        ``{ ` `            ``if` `(n % i == ``0``)  ` `            ``{ ` `                ``return` `true``; ` `            ``} ` `        ``} ` `        ``return` `false``; ` `    ``} ` ` `  `    ``// Effectively calculate (x^y) modulo mod  ` `    ``static` `int` `power(``int` `x, ``int` `y, ``int` `mod) ` `    ``{ ` ` `  `        ``// Initialize result  ` `        ``int` `res = ``1``; ` ` `  `        ``while` `(y != ``0``)  ` `        ``{ ` ` `  `            ``// If power is odd, ` `            ``// then update the answer  ` `            ``if` `((y & ``1``) == ``1``)  ` `            ``{ ` `                ``res = (res * x) % mod; ` `            ``} ` ` `  `            ``// Square the number and reduce  ` `            ``// the power to its half  ` `            ``y = y >> ``1``; ` `            ``x = (x * x) % mod; ` `        ``} ` ` `  `        ``// Return the result  ` `        ``return` `res; ` `    ``} ` ` `  `    ``// Function to check for Fermat Pseudoprime  ` `    ``static` `int` `Check(``int` `n, ``int` `a)  ` `    ``{ ` ` `  `        ``// If it is composite and  ` `        ``// satisfy Fermat criterion  ` `        ``if` `(a > ``1` `&& checkcomposite(n) ` `                ``&& power(a, n - ``1``, n) == ``1``)  ` `        ``{ ` `            ``return` `1``; ` `        ``} ` ` `  `        ``// Else return 0  ` `        ``return` `0``; ` `    ``} ` ` `  `    ``// Driver Code ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `N = ``645``; ` `        ``int` `a = ``2``; ` ` `  `        ``// Function call  ` `        ``System.out.println(Check(N, a)); ` `    ``} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

## Python3

 `# Pyhton 3 program to check if N is Fermat pseudoprime ` `# to the base A or not ` ` `  `from` `math ``import` `sqrt ` ` `  `# Function to check if the given number is composite ` `def` `checkcomposite(n): ` `     `  `    ``# Check if there is any divisor of n less than sqrt(n) ` `    ``for` `i ``in` `range``(``2``,``int``(sqrt(n))``+``1``,``1``): ` `        ``if` `(n ``%` `i ``=``=` `0``): ` `            ``return` `1` `    ``return` `0` ` `  `# Effectively calculate (x^y) modulo mod ` `def` `power(x, y, mod): ` `    ``# Initialize result ` `    ``res ``=` `1` ` `  `    ``while` `(y): ` `        ``# If power is odd, then update the answer ` `        ``if` `(y & ``1``): ` `            ``res ``=` `(res ``*` `x) ``%` `mod ` ` `  `        ``# Square the number and reduce ` `        ``# the power to its half ` `        ``y ``=` `y >> ``1` `        ``x ``=` `(x ``*` `x) ``%` `mod ` ` `  `    ``# Return the result ` `    ``return` `res ` ` `  `# Function to check for Fermat Pseudoprime ` `def` `Check(n,a): ` `    ``# If it is composite and satisfy Fermat criterion ` `    ``if` `(a>``1` `and` `checkcomposite(n) ``and` `power(a, n ``-` `1``, n) ``=``=` `1``): ` `        ``return` `1` ` `  `    ``# Else return 0 ` `    ``return` `0` ` `  `# Driver code ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``N ``=` `645` `    ``a ``=` `2` ` `  `    ``# Function call ` `    ``print``(Check(N, a)) ` ` `  `# This code is contributed by ` `# Surendra_Gangwar `

## C#

 `// C# program to check if N is Fermat pseudoprime  ` `// to the base A or not  ` `using` `System; ` ` `  `class` `GFG ` `{ ` `     `  `    ``// Function to check if  ` `    ``// the given number is composite  ` `    ``static` `bool` `checkcomposite(``int` `n)  ` `    ``{  ` `        ``// Check if there is any divisor of n  ` `        ``// less than sqrt(n)  ` `        ``for` `(``int` `i = 2; i <= Math.Sqrt(n); i++)  ` `        ``{  ` `            ``if` `(n % i == 0)  ` `                ``return` `true``;  ` `        ``}  ` `        ``return` `false``;  ` `    ``}  ` `     `  `    ``// Effectively calculate (x^y) modulo mod  ` `    ``static` `int` `power(``int` `x, ``int` `y, ``int` `mod)  ` `    ``{  ` `     `  `        ``// Initialize result  ` `        ``int` `res = 1;  ` `     `  `        ``while` `(y != 0)  ` `        ``{  ` `     `  `            ``// If power is odd, then update the answer  ` `            ``if` `((y & 1) == 1)  ` `                ``res = (res * x) % mod;  ` `     `  `            ``// Square the number and reduce  ` `            ``// the power to its half  ` `            ``y = y >> 1;  ` `            ``x = (x * x) % mod;  ` `        ``}  ` `     `  `        ``// Return the result  ` `        ``return` `res;  ` `    ``}  ` `     `  `    ``// Function to check for Fermat Pseudoprime  ` `    ``static` `int` `Check(``int` `n, ``int` `a)  ` `    ``{  ` `     `  `        ``// If it is composite and satisfy Fermat criterion  ` `        ``if` `(a > 1 && checkcomposite(n) &&  ` `                     ``power(a, n - 1, n) == 1)  ` `            ``return` `1;  ` `     `  `        ``// Else return 0  ` `        ``return` `0;  ` `    ``}  ` `     `  `    ``// Driver code  ` `    ``static` `public` `void` `Main () ` `    ``{ ` `        ``int` `N = 645;  ` `        ``int` `a = 2;  ` `     `  `        ``// Function call  ` `        ``Console.WriteLine(Check(N, a));  ` `    ``} ` `} ` ` `  `// This code is contributed by AnkitRai01 `

Output:

```1
```

Time Complexity : O(sqrt(N))

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