Exploring the Diophantine Equation \( 2a(a^2 + 3b^2) = (2c)^3 \)

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The Diophantine equation \( 2a(a^2 + 3b^2) = (2c)^3 \) is a mathematical challenge where we seek positive integer solutions \( (a, b, c) \). Interestingly, it has been observed that when \( a = b = c \), the equation holds true. The quest, however, is to determine if there exist solutions where \( a, b, c \) are not all equal.

Python Implementation

To explore this equation in Python, we wrote a straightforward program. The program iterates through possible values of \( a \) and \( b \) up to \( 10^6 \), computing \( c \) and checking if the equation holds. Here’s how the Python code looks:

import math

def find_diophantine_solutions(limit):
    solutions = []
    for a in range(1, limit):
        if a % 100 == 0:
            print(f"Processing a = {a}")

        for b in range(1, limit):
            # Calculate (2c)^3
            right_side = 2 * a * (a**2 + 3 * b**2)
            # Calculate c
            c_cube = right_side // 8  # (2c)^3 / 8 = 2 * a * (a^2 + 3 * b^2)
            c = int(c_cube ** (1/3))
            # Check if (2c)^3 equals to 2a(a^2 + 3b^2)
            if (2 * c)**3 == right_side:
                solutions.append((a, b, c))

    return solutions

limit = 1000000
solutions = find_diophantine_solutions(limit)

print("Solutions found within the limit:")
for solution in solutions:
    print(solution)

This Python script systematically checks each combination of \( a \) and \( b \), calculating \( c \) and verifying if the equation holds true. It also prints the value of \( a \) when it is a multiple of 100, aiding in tracking the progress of the program.

Golang Implementation

Moving to Golang, we first implemented a sequential solution. It iterates similarly through\( a \) and \( b \), checking for solutions. Here’s the sequential Golang code:

package main

import (
    "fmt"
    "math"
    "runtime"
    "sync"
)

func findDiophantineSolutions(limit int) [][]int {
    var solutions [][]int

    for a := 1; a < limit; a++ {
        for b := 1; b < limit; b++ {
            // Calculate right side: 2a(a^2 + 3b^2)
            rightSide := 2 * a * (a*a + 3*b*b)

            // Calculate (2c)^3
            cCube := float64(rightSide) / 8.0
            c := int(math.Pow(cCube, 1.0/3.0))

            // Check if (2c)^3 equals to 2a(a^2 + 3b^2)
            if math.Pow(float64(2*c), 3) == float64(rightSide) {
                solutions = append(solutions, []int{a, b, c})
            }
        }
    }

    return solutions
}

func main() {
    limit := 1000000
    solutions := findDiophantineSolutions(limit)

    fmt.Println("Solutions found within the limit:")
    for _, solution := range solutions {
        fmt.Println(solution)
    }
}

This Golang program mirrors the Python approach, iterating through \( a \) and \( b \), computing \( c \), and verifying the equation. It outputs the solutions found within the specified limit.

Parallelized Golang Implementation

To optimize performance, we parallelized the Golang code using goroutines. Each goroutine handles a distinct range of \( a\) values, leveraging concurrency to speed up the computation. Here’s how the parallelized Golang code looks:

package main

import (
    "fmt"
    "math"
    "runtime"
    "sync"
)

func findDiophantineSolutions(start, end, limit int, results chan<- []int, wg *sync.WaitGroup) {
    defer wg.Done()

    for a := start; a < end; a++ {
        for b := 1; b < limit; b++ {
            // Calculate right side: 2a(a^2 + 3b^2)
            rightSide := 2 * a * (a*a + 3*b*b)

            // Calculate (2c)^3
            cCube := float64(rightSide) / 8.0
            c := int(math.Pow(cCube, 1.0/3.0))

            // Check if (2c)^3 equals to 2a(a^2 + 3b^2)
            if math.Pow(float64(2*c), 3) == float64(rightSide) {
                results <- []int{a, b, c}
            }
        }
    }
}

func main() {
    numCPU := runtime.NumCPU()
    runtime.GOMAXPROCS(numCPU)

    limit := 1000000
    results := make(chan []int)
    var wg sync.WaitGroup

    // Calculate tasks per goroutine
    tasksPerRoutine := limit / numCPU

    // Start goroutines
    for i := 0; i < numCPU; i++ {
        wg.Add(1)
        start := i * tasksPerRoutine
        end := (i + 1) * tasksPerRoutine
        go findDiophantineSolutions(start+1, end+1, limit, results, &wg)
    }

    // Wait for all goroutines to finish
    go func() {
        wg.Wait()
        close(results)
    }()

    // Collect results
    var solutions [][]int
    for solution := range results {
        solutions = append(solutions, solution)
    }

    fmt.Println("Solutions found within the limit:")
    for _, solution := range solutions {
        fmt.Println(solution)
    }
}

In this parallelized Golang code, each goroutine processes a distinct range of \( a \) values. This approach utilizes concurrency to maximize CPU utilization and speed up the computation significantly compared to the sequential implementation.

Conclusion

In conclusion, we explored the Diophantine equation \( 2a(a^2 + 3b^2) = (2c)^3 \) using both Python and Golang. We started with a straightforward Python implementation, then mirrored it in Golang for a sequential approach. Finally, we optimized the Golang code using goroutines to achieve parallel execution, demonstrating how concurrency can dramatically enhance performance when dealing with computationally intensive tasks. Each version of the code served to illustrate different aspects of solving such mathematical equations efficiently and effectively.

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