How to resolve the algorithm Safe primes and unsafe primes step by step in the C# programming language

Published on 12 May 2024 09:40 PM
#C_Sharp

How to resolve the algorithm Safe primes and unsafe primes step by step in the C# programming language

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Problem Statement

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

The provided C# code is designed to investigate and analyze safe and unsafe prime numbers. Here's a detailed explanation of its functionality:

  1. Obtaining Prime Numbers:

    • The code utilizes the Primes method, which generates a sequence of prime numbers up to a specified limit.
    • It uses the Sieve of Eratosthenes algorithm to efficiently find primes. It returns an IEnumerable containing the primes.

  2. Identifying Safe and Unsafe Primes:

    • The code defines two boolean functions: IsSafe and IsUnsafe to classify prime numbers as safe or unsafe.
    • A safe prime is a prime number whose half is also a prime number.
    • An unsafe prime is a prime number whose half is not a prime number.
    • These functions check if the prime number's half is present in the set of primes to determine their safety or unsafeness.

  3. Main Method:

    • The Main method is the entry point of the program.
    • It first generates a set of primes up to 10,000,000 and converts it to a HashSet for efficient lookups.
    • It then prints the first 35 safe primes, the number of safe primes below 1,000,000 and 10,000,000, and the first 40 unsafe primes.
    • It also counts the number of unsafe primes below 1,000,000 and 10,000,000.

Overall, this code demonstrates the concept of safe and unsafe primes, efficiently generates prime numbers, and provides insights into their distribution within a given range. It combines mathematical knowledge with programming techniques to explore an interesting aspect of number theory.

Source code in the csharp programming language

using static System.Console;
using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;

public static class SafePrimes
{
    public static void Main() {
        HashSet<int> primes = Primes(10_000_000).ToHashSet();
        WriteLine("First 35 safe primes:");
        WriteLine(string.Join(" ", primes.Where(IsSafe).Take(35)));
        WriteLine($"There are {primes.TakeWhile(p => p < 1_000_000).Count(IsSafe):n0} safe primes below {1_000_000:n0}");
        WriteLine($"There are {primes.TakeWhile(p => p < 10_000_000).Count(IsSafe):n0} safe primes below {10_000_000:n0}");
        WriteLine("First 40 unsafe primes:");
        WriteLine(string.Join(" ", primes.Where(IsUnsafe).Take(40)));
        WriteLine($"There are {primes.TakeWhile(p => p < 1_000_000).Count(IsUnsafe):n0} unsafe primes below {1_000_000:n0}");
        WriteLine($"There are {primes.TakeWhile(p => p < 10_000_000).Count(IsUnsafe):n0} unsafe primes below {10_000_000:n0}");

        bool IsSafe(int prime) => primes.Contains(prime / 2);
        bool IsUnsafe(int prime) => !primes.Contains(prime / 2);
    }

    //Method from maths library
    static IEnumerable<int> Primes(int bound) {
        if (bound < 2) yield break;
        yield return 2;

        BitArray composite = new BitArray((bound - 1) / 2);
        int limit = ((int)(Math.Sqrt(bound)) - 1) / 2;
        for (int i = 0; i < limit; i++) {
            if (composite[i]) continue;
            int prime = 2 * i + 3;
            yield return prime;
            for (int j = (prime * prime - 2) / 2; j < composite.Count; j += prime) composite[j] = true;
        }
        for (int i = limit; i < composite.Count; i++) {
            if (!composite[i]) yield return 2 * i + 3;
        }
    }

}

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