Step by Step solution about How to resolve the algorithm Descending primes step by step in the C# programming language

The source code provides a program that displays a list of descending prime numbers and counts them while generating. It uses a modified Sieve of Eratosthenes algorithm to efficiently generate prime numbers.

Key implementation details:

1. ispr() Function: This function checks if a given number n is prime. It follows the common prime checking logic:

   • If n is even (except for 2), it's not prime.
   • If n is greater than 2, it checks for divisibility by odd numbers from 3 to the square root of n. If any divisor is found, it returns false. Otherwise, it returns true.

2. Main Logic:

   • The program maintains an array ps to store the set of prime numbers and uses another array nxt to generate the next set of potential prime numbers.
   • It initializes ps with the first few prime numbers (1-9).
   • The outermost while loop continues generating prime numbers until no more candidates can be found.
   • Inside the loop, it iterates through the current prime numbers in ps and filters out the ones that are prime.
   • For each prime a, it appends numbers to nxt that are multiples of a shifted by 10 (e.g., 20, 30, 40, ...). These numbers are placed in the unused slots at the end of nxt.
   • After processing all the primes in ps, it checks if nxt has any unused slots. If it does, it means no more primes can be generated, and the loop breaks. Otherwise, nxt is copied into ps, and the process repeats with the expanded set of potential prime numbers.

3. Output:

   • The program prints prime numbers in descending order, formatted with 8 characters per number.
   • It also counts and displays the total number of descending primes found during the execution.

using System;

class Program {

  static bool ispr(uint n) {
    if ((n & 1) == 0 || n < 2) return n == 2;
    for (uint j = 3; j * j <= n; j += 2)
      if (n % j == 0) return false; return true; }

  static void Main(string[] args) {
    uint c = 0; int nc;
    var ps = new uint[]{ 1, 2, 3, 4, 5, 6, 7, 8, 9 };
    var nxt = new uint[128];
    while (true) {
      nc = 0;
      foreach (var a in ps) {
        if (ispr(a))
          Console.Write("{0,8}{1}", a, ++c % 5 == 0 ? "\n" : " ");
        for (uint b = a * 10, l = a % 10 + b++; b < l; b++)
          nxt[nc++] = b;
      }
      if (nc > 1) {
        Array.Resize (ref ps, nc); Array.Copy(nxt, ps, nc); }
      else break;
    }
    Console.WriteLine("\n{0} descending primes found", c);
  }
}