Question

Find the smallest strong pseudoprime to the base 3 in the interval [700, 800]. Kindly show all trials without using any programming language code.

Answer 1

code in cpp:

#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 %2)== 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(strong) 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;
}

int main()
{

   int a=3;//to the base which we want to find
  
// Function call
   for(int i=700;i<=800;i++){
       if(Check(i, a)){
           cout<<i<<endl;
           break;
       }
   }
  

output:-

703

   return 0;
}