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

The given code in C# implements a program that finds and displays the "long primes" within a specified range and counts their occurrences within certain limits. An integer n is considered a "long prime" if the period of its decimal expansion is equal to n-1.

Here's a detailed explanation of the code:

1. Main Method:

   • The Main method serves as the program's entry point.
   • It generates a sequence of prime numbers up to 64000 using the Primes method and filters out the first prime (2) using Skip(1).
   • The primes are then filtered to include only those where the period of their decimal expansion is equal to the prime number minus 1 (Where(p => Period(p) == p - 1)).
   • The sequence is finally extended by adding a prime (99999) as a sentinel value.

2. Console Output:

   • The Main method prints the first 500 primes from the generated sequence.
   • It also counts the number of long primes below different limits and prints this count at those limits (initially set to 500, doubling each time).

3. Period Method:

   • This method calculates the period of the decimal expansion of an integer n.
   • It starts with an initial remainder r of 1 and iterates until r becomes equal to the initial remainder again.
   • The number of iterations required to reach this point is the period.

4. SomePrimeGenerator Class:

   • This class contains a static method Primes that generates prime numbers up to a specified limit.
   • It uses the Sieve of Eratosthenes algorithm to mark non-prime numbers within an array flags.
   • Prime numbers are then yielded from the sequence.

5. Program Output:

   • The program prints the first 500 long primes, along with the counts of long primes below various limits (such as 500, 1000, 2000, etc.).

In summary, this program finds and counts "long primes" (prime numbers where the period of their decimal expansion equals the prime number minus 1) within a specified range. It also prints the count of these primes below certain limits and provides the first 500 long primes found.

using System;
using System.Collections.Generic;
using System.Linq;

public static class LongPrimes
{
    public static void Main() {
        var primes = SomePrimeGenerator.Primes(64000).Skip(1).Where(p => Period(p) == p - 1).Append(99999);
        Console.WriteLine(string.Join(" ", primes.TakeWhile(p => p <= 500)));
        int count = 0, limit = 500;
        foreach (int prime in primes) {
            if (prime > limit) {
                Console.WriteLine($"There are {count} long primes below {limit}");
                limit *= 2;
            }
            count++;
        }

        int Period(int n) {
            int r = 1, rr;
            for (int i = 0; i <= n; i++) r = 10 * r % n;
            rr = r;
            for (int period = 1;; period++) {
                r = (10 * r) % n;
                if (r == rr) return period;
            }
        }
    }

}

static class SomePrimeGenerator {

    public static IEnumerable<int> Primes(int lim) {
        bool [] flags = new bool[lim + 1]; int j = 2;
        for (int d = 3, sq = 4; sq <= lim; j++, sq += d += 2)
            if (!flags[j]) {
                yield return j; for (int k = sq; k <= lim; k += j)
                    flags[k] = true;
            }
        for (; j<= lim; j++) if (!flags[j]) yield return j;
    }
}