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

This C++ program calculates and displays the first 10 primorial numbers and the length of primorial numbers up to a specific index.

#include <gmpxx.h> // For GMP++ integer operations
#include <primesieve.hpp> // For the primesieve library

#include <cstdint> // For uint64_t
#include <iomanip> // For setw()
#include <iostream> // For cout

size_t digits(const mpz_class& n) { return n.get_str().length(); }

mpz_class primorial(unsigned int n) {
   mpz_class p;
   mpz_primorial_ui(p.get_mpz_t(), n); // Calculate the nth primorial number
   return p;
}

int main() {
   uint64_t index = 0;
   primesieve::iterator pi; // Prime sieve iterator

   // Display the first 10 primorial numbers
   std::cout << "First 10 primorial numbers:\n";
   for (mpz_class pn = 1; index < 10; ++index) {  
       unsigned int prime = pi.next_prime(); // Get the next prime number
       std::cout << index << ": " << pn << '\n';
       pn *= prime; // Multiply the primorial number by the next prime
   }

   // Output the length of the primorial number for various indices
   std::cout << "\nLength of primorial number whose index is:\n";
   for (uint64_t power = 10; power <= 1000000; power *= 10) {
       uint64_t prime = primesieve::nth_prime(power); // Get the prime number at the given index
       std::cout << std::setw(7) << power << ": " << digits(primorial(prime)) << '\n';
   }
   return 0;
}

Here's how the program works:

1. Calculating Primorials: The primorial function calculates the nth primorial number using GMP++. Primorial numbers are the product of the first n prime numbers. The function takes an integer n as input and returns the nth primorial number.

2. Finding the First 10 Primorial Numbers: The program initializes an iterator pi for the primesieve library to generate prime numbers. It iterates through the first 10 primes, calculates the corresponding primorial numbers, and displays their indices and values.

3. Determining the Length of Primorial Numbers: The program calculates the length of primorial numbers for various indices represented by powers of 10 (from 10 to 1,000,000). For each power, it finds the corresponding prime number using primesieve::nth_prime and then calculates the length of the primorial number at that index using the digits function.

By running the program, you will get the first 10 primorial numbers and the length of primorial numbers for various indices. This demonstrates the use of GMP++ for large integer calculations and the primesieve library for efficient prime number generation.

#include <gmpxx.h>
#include <primesieve.hpp>

#include <cstdint>
#include <iomanip>
#include <iostream>

size_t digits(const mpz_class& n) { return n.get_str().length(); }

mpz_class primorial(unsigned int n) {
    mpz_class p;
    mpz_primorial_ui(p.get_mpz_t(), n);
    return p;
}

int main() {
    uint64_t index = 0;
    primesieve::iterator pi;
    std::cout << "First 10 primorial numbers:\n";
    for (mpz_class pn = 1; index < 10; ++index) {
        unsigned int prime = pi.next_prime();
        std::cout << index << ": " << pn << '\n';
        pn *= prime;
    }
    std::cout << "\nLength of primorial number whose index is:\n";
    for (uint64_t power = 10; power <= 1000000; power *= 10) {
        uint64_t prime = primesieve::nth_prime(power);
        std::cout << std::setw(7) << power << ": "
                  << digits(primorial(prime)) << '\n';
    }
    return 0;
}