Stormer Numbers

For N to be a stormer number, the highest prime factor of the expression N^2+1 must be greater than or equal to 2*N and it should be a positive integer.

For example, 4 is a stormer number. Since 4*4+1=17 has the greatest prime factor 17 itself which is greater than 8 i.e. 2*4.

But 3 is not a stormer number because 3*3+1=10. The greatest prime factor of 10 is 5 which is less than 6 i.e. 2*3.

In this problem, we are given a positive integer N and our goal is to print the first N stormer.

INPUT: 4

OUTPUT: 1 2 4 5

Here are the first 4 stormer numbers. 3 is not a stormer number that’s why it is not included.

Algorithm

• Find the biggest prime factor of the no (N^2+1) and store it in any variable.

• Check if the prime factor is larger than or equal to 2*N.

• If it satisfies the condition hence it is a stormer number.

• Print all the stormer numbers until i is less than or equal to N.

Approach

To implement the above algorithm in our code, we need to make two functions. First, to find out the highest prime factor for each case and second to check if it is greater than or equal to 2*N and keep printing the number if it is a stormer number simultaneously.

For finding out the highest prime factor of expression (n^2+1) for every number n −

• We will divide the number by 2 until it gives remainder 0 and store 2 in primemax.

• Now, n must be odd at this point so we will iterate in a for loop for odd integers only from i=3 to the square root of n.

• Now store i in primemax and divide n by i while i divides n. When i fails to divide n then raise it by 2 and carry on.

• If n is a prime number and greater than 2, n cannot become 1 by the previous two steps thus we will store n in primemax and return primemax.

The next function will be to check if the number is a stormer number or not. If it is, we will print it.

• We will declare a variable temp as 0 to count first N stormer numbers.

• Iterate in a for loop from i=1 until temp is less than N.

• Check if (i*i+1) has the greatest prime factor which is greater than or equal to 2*i. If the above condition is true print i and increase temp by 1.

Example

Below is the implementation of above approach in C++ −

#include <iostream>
#include <bits/stdc++.h>

using namespace std;

int prime_factor(int n){ //for finding the maximum prime factor
   int primemax=0;
   while(n%2==0){ //if n is divided by 2 than we store 2 in primemax 
      primemax=2;
      n=n/2;
   }
   for(int i=3;i<=sqrt(n);i+=2){ // this is only for odd number 
      while(n%i==0){
         primemax=i;
         n=n/i;
      }
   }
   if(n>2){ // when n is prime number and greater than 2 
      primemax=n;
   }
   return primemax;
}
void stormer_number(int n){  //function to print first n stormer numbers
   int temp=0; // for counting the stormer number 
   for(int i=1;temp<n;i++){  // for iterating the all number 
      int check=(i*i)+1;  // for storing the (i*i)+1 value in check
      if(prime_factor(check)>=(2*i)){ //for checking the number if maximum prime is greater or equal to 2*i
         cout<<i<<" ";
         temp++;
      }
   }
}
int main(){
   int n=9;
   stormer_number(n);
   
   return 0;
}

Output

1 2 4 5 6 9 10 11 12

Conclusion

In this article, we have tried to solve the problem to print first N stormer numbers.

We have also learnt to calculate the prime factor of a number. I hope this article helps to clear all your doubts regarding the problem.