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
Approach: The approach is to check the below conditions:
- Check if A > 1.
- Check if N is a composite number.
- Check if N divides AN-1 – 1.
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++ program to check if N is Fermat pseudoprime // to the base A or not #include <bits/stdc++.h> 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 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 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# 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 |
<script> // Javascript program to check if // N is Fermat pseudoprime // to the base A or not // Function to check if the given // number is composite function checkcomposite(n)
{ // Check if there is any divisor
// of n less than sqrt(n)
for (let i = 2; i <= Math.sqrt(n); i++)
{
if (n % i == 0)
return 1;
}
return 0;
} // Effectively calculate (x^y) modulo mod function power(x, y, mod)
{ // Initialize result
let 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 function Check(n, 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 let N = 645;
let a = 2;
// Function call
document.write(Check(N, a));
</script> |
1
Time Complexity : O(sqrt(N))
Auxiliary Space: O(1)