Ramanujan Third Problem

A person born here would revolutionize mathematics forever. Would give a different perspective to the way we interpret infinite sums and series. Would come up with novel ways to think about hard mathematical problems such as the Partition Problem. A person born here, raised by his mother alone, with no higher education, no computers to assist him, just thru hard relentless work, would accomplish so much in such a short period of time.

I continue to be amazed by the work and life of Srinivasa Ramanujan. I think one of his most important contributions was to come up with a formula for the Partition Problem:

What fascinates me the most is that mathematicians like him never had the power of computers at their disposal. One of the anecdotes that is very interesting from Ramanujan happened when he was already very ill in England and was visited by G. H. Hardy who rode a taxi numbered 1729. When he mentioned this to Ramanujan, he hoped that this wasn't a sign of bad luck given that "1729 looked like such a dull, ordinary number", for which Ramanujan famously replied:

No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.

Imagine Ramanujan with his hands on a computer. He could even extend this statement by looking for numbers that match the following criteria:

"The smallest number expressible as the sum of N powers of P in K different ways"

For example, for P = 9, N = 9 and K = 2, you'll find this number: 458143414265583

458143414265583 = 1^9 + 2^9 + 4^9 + 14^9 + 16^9 + 18^9 + 20^9 + 37^9 + 41^9
458143414265583 = 1^9 + 2^9 + 5^9 + 8^9 + 15^9 + 22^9 + 25^9 + 33^9 + 42^9

Code is down below, cheers, ACC.

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Numerics;

namespace Ramanujan1729
{
    internal class Program
    {
        static void Main(string[] args)
        {
            string MAXLEN = "1000000000000000";

            if (args.Length == 1 && args[0].Equals("-all"))
            {
                for (int p = 3; p <= 10; p++)
                {
                    for (int n = 2; n <= 10; n++)
                    {
                        bool breakPoint = false;
                        for (int c = 2; c <= 5; c++)
                        {
                            try
                            {
                                Console.WriteLine("p={0}, n={1}, c={2}", p, n, c);
                                if (!SumOfPowers(0,
                                                1,
                                                "",
                                                p,
                                                n,
                                                c,
                                                new Hashtable(),
                                                BigInteger.Parse(MAXLEN)))
                                {
                                    break;
                                }
                                Console.WriteLine();
                            }
                            catch (OutOfMemoryException e)
                            {
                                Console.WriteLine("Error: {0}", e.Message);
                                Console.WriteLine("Continuing...");
                                breakPoint = true;
                                break;
                            }
                        }

                        if (breakPoint) break;
                    }
                }
                return;
            }
            else if (args.Length != 3)
            {
                Console.WriteLine("Usage: Ramanujan1729.exe   ");
                Console.WriteLine("Example: Ramanujan1729.exe 3 2 2");
                return;
            }
            SumOfPowers(0, 
                        1, 
                        "", 
                        Int32.Parse(args[0]), 
                        Int32.Parse(args[1]), 
                        Int32.Parse(args[2]), 
                        new Hashtable(), 
                        BigInteger.Parse(MAXLEN));
        }

        static bool SumOfPowers(BigInteger sum,
                                BigInteger startNumber,
                                string numbers,
                                int power,
                                int n,
                                int uniqueComb,
                                Hashtable htSum,
                                BigInteger MAX)
        {
            if (sum > MAX || htSum.Count >= 10000000) return false;

            if (n == 0)
            {
                if (!htSum.ContainsKey(sum)) htSum.Add(sum, new Hashtable());
                Hashtable combinations = (Hashtable)htSum[sum];
                if (!combinations.ContainsKey(numbers.Trim())) combinations.Add(numbers.Trim(), true);

                if (combinations.Count == uniqueComb)
                {
                    Console.WriteLine("Found: n = {0}", sum);
                    foreach (string num in combinations.Keys)
                    {
                        Console.WriteLine(num);
                    }
                    return true;
                }
                return false;
            }

            for (BigInteger i = startNumber, p = BigInteger.Pow(i, power); p <= MAX; i++, p = BigInteger.Pow(i, power))
            {
                if (SumOfPowers(sum + p, i + 1, numbers + " " + i.ToString(), power, n - 1, uniqueComb, htSum, MAX)) return true;
            }

            return false;
        }
    }
}