Step by Step solution about How to resolve the algorithm Carmichael 3 strong pseudoprimes step by step in the C programming language

This C code finds Carmichael numbers that satisfy specific mathematical properties. Carmichael numbers are composite numbers that behave like prime numbers under certain modular arithmetic operations. The code uses a nested loop to generate and test Carmichael numbers within a specified range.

Here's a detailed breakdown of the code:

1. Header File and Macro:

   • The code includes the <stdio.h> header file for input and output.
   • It defines a macro mod(n, m) that calculates the remainder of n divided by m while ensuring that the result is positive, even for negative n values.

2. is_prime() Function:

   • This function checks if a given positive integer n is prime or not.
   • It uses a series of conditions to efficiently handle small values and prime numbers that are multiples of 2 or 3.
   • For larger numbers, it uses a loop to check for divisibility by potential factors.

3. carmichael3() Function:

   • This function searches for Carmichael numbers with three prime factors p1, p2, and p3 that satisfy certain properties.
   • It iterates through potential values of h3 and d to find triples that meet the conditions outlined in the loop.
   • If it finds a valid triple, it calculates p2 and p3 and checks if they are prime.
   • Finally, it prints the Carmichael number sequence p1, p2, and p3 if they satisfy all the conditions.

4. main() Function:

   • In the main function, the code iterates from p1 = 2 to p1 = 62 and calls the carmichael3() function for each p1.
   • This loop finds and prints Carmichael numbers within this range that satisfy the specified properties.

In summary, this code generates and identifies Carmichael numbers with three prime factors that meet specific mathematical conditions. Carmichael numbers are a curious class of numbers that exhibit properties similar to prime numbers in certain modular arithmetic operations.

#include <stdio.h>

/* C's % operator actually calculates the remainder of a / b so we need a
 * small adjustment so it works as expected for negative values */
#define mod(n,m) ((((n) % (m)) + (m)) % (m))

int is_prime(unsigned int n)
{
    if (n <= 3) {
        return n > 1;
    }
    else if (!(n % 2) || !(n % 3)) {
        return 0;
    }
    else {
        unsigned int i;
        for (i = 5; i*i <= n; i += 6)
            if (!(n % i) || !(n % (i + 2)))
                return 0;
        return 1;
    }
}

void carmichael3(int p1)
{
    if (!is_prime(p1)) return;

    int h3, d, p2, p3;
    for (h3 = 1; h3 < p1; ++h3) {
        for (d = 1; d < h3 + p1; ++d) {
            if ((h3 + p1)*(p1 - 1) % d == 0 && mod(-p1 * p1, h3) == d % h3) {
                p2 = 1 + ((p1 - 1) * (h3 + p1)/d);
                if (!is_prime(p2)) continue;
                p3 = 1 + (p1 * p2 / h3);
                if (!is_prime(p3) || (p2 * p3) % (p1 - 1) != 1) continue;
                printf("%d %d %d\n", p1, p2, p3);
            }
        }
    }
}

int main(void)
{
    int p1;
    for (p1 = 2; p1 < 62; ++p1)
        carmichael3(p1);
    return 0;
}