Number Theory: Sum of Cube

Date: 7/6/96 at 4:42:0
From: Anonymous
Subject: Number Theory: Sum of Cube

Any positive integer n can be expressed as a sum of cube.
  ex:  
    567 = 1^3 + 1^3 + .. + 1^3    (where 1^3 is repeated 567 times)
                                  (order is 567)  
   ...but more interestingly this can be expressed as
    567 = 8^3 + 3^3 + 3^3 + 1^3   (order is 4)  
   or
    567 = 7^3 + 6^3 + 2^3         (order is 3)
     
    
  Define the order of n:
     Order of n is equal to the number of term(s) that makes up n.

  Problem: 
     Make an accurate solution that will generate the sum of cube for 
     a given integer n, where the order of n is the least among all  
     the possible answers.

      ex.
         567 of order 3 (567 = 7^3 + 6^3 + 2^3) is the prefered answer
            rather than that of order 4 (567 = 8^3 + 3^3 + 3^3 + 1^3).

Date: 7/12/96 at 17:26:28
From: Doctor Pete
Subject: Re: Number Theory

As far as I am aware, this is an open problem; that is, no known 
solution exists.  Your question is related to Waring's problem.  Let 
g(n) denote the least number of terms required to express any positive 
integer as the sum of (n)th powers.  It has been shown that g(3) = 9; 
i.e., the least number of cubes needed to express every positive 
integer as a sum of cubes, is 9. There are only two integers (23 and 
239, I think) that require 9 cubes; every other number can be 
expressed as the sum of at most 8 cubes.  Now, we define G(n) in a 
similar manner to g(n):  Let G(n) denote the least number of terms 
needed to express any *sufficiently large* positive integer as the sum 
of (n)th powers.  So only finitely many positive integers will require 
more than G(n) terms to be expressed.  The current bounds on G(3) is

     4 <= G(3) <= 7,

so no exact value is known.  I believe that if one were able to find a
formula or algorithm to determine the minimal expression of any 
positive integer as the sum of cubes, G(3) could be determined 
exactly; but this has not been done.  But I am not absolutely certain 
of this

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/