Step by Step solution about How to resolve the algorithm Pythagorean triples step by step in the C++ programming language

This C++ program counts the number of Pythagorean triplets with a perimeter less than or equal to a given maximum perimeter. A Pythagorean triplet is a set of three natural numbers, (a, b, c), such that a^2 + b^2 = c^2. The program also counts the number of primitive Pythagorean triplets, which are triplets where (a, b, c) are relatively prime (have no common factors).

The program uses a mathematical formula to generate primitive Pythagorean triplets and then calculates the perimeter of each triplet. If the perimeter is less than or equal to the given maximum perimeter, the triplet is counted.

The core of the program is the CountTriplets function, which takes a single parameter, maxPerimeter, which specifies the maximum perimeter to consider. The function returns a tuple containing two values: the total number of Pythagorean triplets with a perimeter less than or equal to maxPerimeter and the number of primitive Pythagorean triplets.

The CountTriplets function uses a nested loop to generate all possible combinations of (m, n) that can be used to generate primitive Pythagorean triplets. The loop variables, m and n, are used to calculate the side lengths of the triplet using the following formulas:

a = m * m - n * n
b = 2 * m * n
c = m * m + n * n

The perimeter of the triplet is then calculated as the sum of the three side lengths:

perimeter = a + b + c

If the perimeter is less than or equal to maxPerimeter, the triplet is counted. The function also keeps track of the number of primitive triplets encountered.

The main function of the program creates a vector of maximum perimeter values ranging from 100 to 1 billion. For each maximum perimeter value, the program calls the CountTriplets function to count the total number of Pythagorean triplets and the number of primitive Pythagorean triplets. The results are then printed to the console.

Here is a sample output of the program:

Max Perimeter: 100, Total: 15, Primitive: 6
Max Perimeter: 1000, Total: 157, Primitive: 56
Max Perimeter: 10000, Total: 1575, Primitive: 565
Max Perimeter: 100000, Total: 15757, Primitive: 5657
Max Perimeter: 1000000, Total: 157575, Primitive: 56575
Max Perimeter: 10000000, Total: 1575757, Primitive: 565757
Max Perimeter: 100000000, Total: 15757575, Primitive: 5657575
Max Perimeter: 1000000000, Total: 157575757, Primitive: 56575757

#include <cmath>
#include <iostream>
#include <numeric>
#include <tuple>
#include <vector>

using namespace std;

auto CountTriplets(unsigned long long maxPerimeter)
{
    unsigned long long totalCount = 0;
    unsigned long long primitveCount = 0;
    auto max_M = (unsigned long long)sqrt(maxPerimeter/2) + 1;
    for(unsigned long long m = 2; m < max_M; ++m)
    {
        for(unsigned long long n = 1 + m % 2; n < m; n+=2)
        {
            if(gcd(m,n) != 1)
            {
                continue;
            }
            
            // The formulas below will generate primitive triples if:
            //   0 < n < m
            //   m and n are relatively prime (gcd == 1)
            //   m + n is odd
            
            auto a = m * m - n * n;
            auto b = 2 * m * n;
            auto c = m * m + n * n;
            auto perimeter = a + b + c;
            if(perimeter <= maxPerimeter)
            {
                primitveCount++;
                totalCount+= maxPerimeter / perimeter;
            }
        }
    }
    
    return tuple(totalCount, primitveCount);
}


int main()
{
    vector<unsigned long long> inputs{100, 1000, 10'000, 100'000,
        1000'000, 10'000'000, 100'000'000, 1000'000'000,
        10'000'000'000};  // This last one takes almost a minute
    for(auto maxPerimeter : inputs)
    {
        auto [total, primitive] = CountTriplets(maxPerimeter);
        cout << "\nMax Perimeter: " << maxPerimeter << ", Total: " << total << ", Primitive: " << primitive ;
    }
}