Step by Step solution about How to resolve the algorithm Primorial numbers step by step in the C# programming language

The provided C# code is designed to generate prime numbers up to a specified limit and then calculate the "primorial" of those primes. The primorial of a number is the product of all prime numbers less than or equal to that number.

Here's a detailed explanation of the code:

1. Generating Prime Numbers (gp() Method):

   • The gp() method takes an integer n as input and returns an array of prime numbers less than n.
   • It uses the Sieve of Eratosthenes algorithm to efficiently find prime numbers.
   • The method initializes a boolean array c to mark numbers as composite and an integer array r to store the prime numbers.
   • It starts by marking all multiples of 9 (except 2) as composite.
   • Then, it iterates through the remaining numbers, starting from 5, and marks their multiples as composite.
   • The method adds the unmarked numbers to the r array and returns it.

2. Main Method:

   • The Main method is the entry point of the program.
   • It starts a stopwatch (sw) to measure the time taken for various operations.

3. Generating Prime Numbers and Primorial Calculations:

   • The program calls the gp method to generate an array of prime numbers up to a specified limit (15485864) and stores the result in the res array.
   • It then calculates the time taken to generate the primes (gpt).

4. Nth Primorial Calculation:

   • The program initializes an array s to store a table of values.
   • It iterates through the prime numbers (res) and calculates the primorial for each index (i).
   • It also calculates the number of digits in the primorial (y).

5. Time Measurements and Tabulation:

   • The program measures the time taken to calculate the primorials for various limits (1, 10, 100, 1000, 10000, 100000, 1000000) and records the results in the s array.

6. Console Output:

   • The program prints the table of values, along with the total time taken for both prime generation and primorial calculations.

Overall, this code demonstrates the use of the Sieve of Eratosthenes for prime number generation and calculates the primorials of those primes. It also measures and reports the time taken for these operations.

using System;

class Program {

    static int l;

    static int[] gp(int n) {
        var c = new bool[n]; var r = new int[(int)(1.28 * n)];
        l = 0; r[l++] = 2; r[l++] = 3; int j, d, lim = (int)Math.Sqrt(n);
        for (int i = 9; i < n; i += 6) c[i] = true; 
        for (j = 5, d = 4; j < lim; j += (d = 6 - d)) if (!c[j]) { r[l++] = j;
            for (int k = j * j, ki = j << 1; k < n; k += ki) c[k] = true; }
        for ( ; j < n; j += (d = 6 - d)) if (!c[j]) r[l++] = j; return r; }

    static void Main(string[] args) {
        var sw = System.Diagnostics.Stopwatch.StartNew();
        var res = gp(15485864); sw.Stop();
        double gpt = sw.Elapsed.TotalMilliseconds, tt;
        var s = new string[19]; int si = 0;
        s[si++] = String.Format("primes gen time: {0} ms", gpt); sw.Restart();
        s[si++] = "    Nth Primorial";
        double y = 0; int x = 1, i = 0, lmt = 10;
        s[si++] = String.Format("{0,7} {1}", 0, x);
        while (true) {
            if (i < 9) s[si++] = String.Format("{0,7} {1}", i + 1, x *= res[i]);
            if (i == 8) s[si++] = "    Nth Digits     Time";
            y += Math.Log10(res[i]);
            if (++i == lmt) {
                s[si++] = String.Format("{0,7} {1,-7}  {2,7} ms", lmt, 1 + (int)y,
                    sw.Elapsed.TotalMilliseconds);
                if ((lmt *= 10) > (int)1e6) break;    }    }
        sw.Stop();
        Console.WriteLine("{0}\n     Tabulation: {1} ms", string.Join("\n", s),
            tt = sw.Elapsed.TotalMilliseconds);
        Console.Write("          Total:{0} ms", gpt + tt);    }    }