How to find all taxicab numbers less than N?

Question

A taxicab number is an integer that can be expressed as the sum of two cubes of integers in two different ways: a^3+b^3 = c^3+d^3. Design an algorithm to find all t

Answer 1

version1 uses List and sorting
O(n^2*logn) time and O(n^2) space

    public static void Taxicab1(int n)
    {
        // O(n^2) time and O(n^2) space
        var list = new List();
        for (int i = 1; i <= n; i++)
        {
            for (int j = i; j <= n; j++)
            {
                list.Add(i * i * i + j * j * j);
            }
        }

        // O(n^2*log(n^2)) time
        list.Sort();

        // O(n^2) time
        int prev = -1;
        foreach (var next in list)
        {
            if (prev == next)
            {
                Console.WriteLine(prev);
            }
            prev = next;
        }
    }

version2 uses HashSet
O(n^2) time and O(n^2) space

    public static void Taxicab2(int n)
    {
        // O(n^2) time and O(n^2) space
        var set = new HashSet();
        for (int i = 1; i <= n; i++)
        {
            for (int j = i; j <= n; j++)
            {
                int x = i * i * i + j * j * j;
                if (!set.Add(x))
                {
                    Console.WriteLine(x);
                }
            }
        }
    }

version3 uses min oriented Priority Queue
O(n^2*logn) time and O(n) space

    public static void Taxicab3(int n)
    {
        // O(n) time and O(n) space
        var pq = new MinPQ();
        for (int i = 1; i <= n; i++)
        {
            pq.Push(new SumOfCubes(i, i));
        }

        // O(n^2*logn) time
        var sentinel = new SumOfCubes(0, 0);
        while (pq.Count > 0)
        {
            var current = pq.Pop();

            if (current.Result == sentinel.Result)
                Console.WriteLine($"{sentinel.A}^3+{sentinel.B}^3 = {current.A}^3+{current.B}^3 = {current.Result}");

            if (current.B <= n)
                pq.Push(new SumOfCubes(current.A, current.B + 1));

            sentinel = current;
        }
    }

where SummOfCubes

public class SumOfCubes : IComparable
{
    public int A { get; private set; }
    public int B { get; private set; }
    public int Result { get; private set; }

    public SumOfCubes(int a, int b)
    {
        A = a;
        B = b;
        Result = a * a * a + b * b * b;
    }

    public int CompareTo(SumOfCubes other)
    {
        return Result.CompareTo(other.Result);
    }
}

github