Step by Step solution about How to resolve the algorithm Sequence of primes by trial division step by step in the C++ programming language

The provided C++ code is designed to find and display all prime numbers within a specified range. Here's a step-by-step explanation :

1. Header Inclusions: • <math.h>: Includes mathematical functions. • <iostream> and <iomanip>: Used for input/output operations and manipulating output formatting.

2. isPrime Function: • This function determines whether a given unsigned integer u is prime. • The function first checks if u is less than 4. If u is 1, 2, or 3, it returns true. • If u is even (divisible by 2), it returns false. If u is divisible by 3, it also returns false. • It then calculates the square root of u, represented by q, and a counter c is initialized to 5. • The function checks if u is divisible by either c or c + 2. If it is, it returns false. • It increments c by 6 and continues checking until c is greater than or equal to q. • If none of the checks find factors, the function returns true, indicating that u is prime.

3. main Function: • Variables: • mx is set to 100000000, representing the upper limit of the prime number search range. • wid is calculated based on the logarithm of mx, plus 1. This determines the width used when displaying prime numbers. • Prime Number Search: • The loop starts with u set to 3, as 2 is already known to be prime. • It iterates through odd numbers up to mx. • For each u, it calls the isPrime function to check if it's prime. If it is, the prime number is displayed, and the count p is incremented. • The loop continues until u reaches the given limit. • Output: • The program prints the prime numbers found within the specified range, with each number right-aligned within the calculated width wid. • It also displays the total number of prime numbers found.

#include <math.h>
#include <iostream>
#include <iomanip>

bool isPrime( unsigned u ) {
    if( u < 4 ) return u > 1;
    if( /*!( u % 2 ) ||*/ !( u % 3 ) ) return false;

    unsigned q = static_cast<unsigned>( sqrt( static_cast<long double>( u ) ) ),
             c = 5;
    while( c <= q ) {
        if( !( u % c ) || !( u % ( c + 2 ) ) ) return false;
        c += 6;
    }
    return true;
}
int main( int argc, char* argv[] )
{
    unsigned mx = 100000000,
             wid = static_cast<unsigned>( log10( static_cast<long double>( mx ) ) ) + 1;

    std::cout << "[" << std::setw( wid ) << 2 << " ";
    unsigned u = 3, p = 1; // <- start computing from 3
    while( u < mx ) {
        if( isPrime( u ) ) { std::cout << std::setw( wid ) << u << " "; p++; }
        u += 2;
    }
    std::cout << "]\n\n Found " << p << " primes.\n\n";
    return 0;
}