Finding a sum of two cubes representation for any Gaussian Integer

 

 

 

I present some methods of obtaining sum of two cubes representations for Gaussian Integers.  The order in which I present the methods does not necessarily indicate the various algorithms’ efficiency, and ease of use.

 

Note: I use MathZoom on most of my MathPage papers.  So, you should be able to click on the MathType equations, and formulas to enlarge them.

 

 

This paper was inspired by problem 30 on p. 82 of [1]:

 

                   

 

‘shew’ is how it is in [1].  Allot of older math texts appear to use this word.

 

It immediately occurred to me that x,y,z are the roots of a reduced cubic equation, and a,b correspond to two numbers that the arbitrary root is split into in order to solve the cubic equation.  That is,

 

 

 

Now, if we can compose an appropriate reduced cubic equation where some number, N, we wish a sum of two cubes representation for is equal to the constant term in (1) above, i.e. G, then, u,v will be numbers that when cubed and added will yield N.

 

First we need to take any two divisors of N, whose product is N. (This paper shows how to find all factors of any Gaussian Integer… Finding all factors of a number)  Any two divisors whose product is N will do.  All such pairs of divisors will yield a sum of two cubes representation of N.

 

What we need are two numbers such that their sum will yield one of the two factors of N taken above, and whose product will yield the other factor of N taken above.  Of course, these would be the roots of a quadratic equation.

 

Then, with the two roots of this quadratic equation, we can form the coefficients of a reduced cubic equation having the roots we desire, from which we can determine the u,v such that u³+v³ = N.  More specifically,

 

 

 

Actually, we don’t necessarily need the coefficients to this reduced cubic equation to find the sum of two cubes representation, but all this makes it easier to understand why this method of finding the sum of two cubes works.

 

The first method I found does require the coefficients to the reduced cubic equation, particularly the coefficient to the linear term.  I thought this method might have some advantages, but it doesn’t appear to work in all cases for reasons I’ve not yet determined.  If anyone has any ideas as to why it doesn’t work in certain cases, particularly with complex Gaussian Integers, please let me know.

 

I’ll first present the method that requires the linear term of the reduced cubic equation above,… saying again, that it doesn’t appear to work in all cases.  Then, I’ll present another method that does appear to work in all cases.

The difference in the two methods is in the choice of pairs of equations to solve simultaneously for u,v.  Below is shown the pair of equations to solve for the first method:

 

 

 

u and v are related in such a manner such that u is equal to (11) taken with the positive sign in the latter portion of the equation, and v is equal to (11) taken with the negative sign.

 

As I’ve said, however, (11) doesn’t appear to yield the correct u,v in all cases for reasons I’ve not yet determined.  So, I moved on to find another pair of equations to solve simultaneously for u,v as is shown here:

 

      

 

The second method has certain advantages as well, particularly that you don’t need to know any of the coefficients of the reduced cubic equation (1).

 

I’ll present a few examples utilizing equations (12) to determine the sum of two cubes for various numbers.  First, I’ll start with 35:

 

 

 

As I indicated, all such pairs of divisors of N whose product is N will yield a sum of two cubes representation:

 

 

 

Finally, I’ll provide one example of determining a sum of two cubes representation of a complex Gaussian Integer.  Even though I fixed the numbers so computation would be easier, the work is quite extensive.  In general, for most complex Gaussian Integers, the work would be even more extensive.

 

 

 

 

As I stated previously, it appears every Gaussian Integer has at least one sum of two cubes representation.  Of course, not every rational integer has a rational integer sum of two cubes representation.  I’ve not demonstrated any method to make any determination of whether or not a rational integer has a rational integer sum of two cubes representation. (This last method outlined I believe will provide a method of making such a determination.)

 

This is a method for determining whether a rational integer has a rational integer sum of two cubes representation, and what that representation is, as presented on p. 148 of [2].  I’ll outline this method here.

 

Given that one wishes to determine if there exists a rational integer sum of two cubes representation for a given rational integer, one first determines the prime factorization of the given rational integer, then proceeds as outlined below:

 

(This example is just the one presented in [2])

 

 

 

I suspect that if formula (15) were used with complex Gaussian integer factors, it would yield other sums of two cubes representations over the Gaussian numbers.

 

In fact, that probably highlights a good point.  These methods can yield sum of two cubes representations for any Gaussian Integer, but that doesn’t imply that the two cubes comprising any of these sums will be Gaussian Integers.  I believe they’ll be Gaussian algebraic numbers, but generally, they won’t be Gaussian Integers.

 

The method I just outlined from [2] is generally probably simpler and easier than the methods I presented earlier.  If the method of [2] works as I suspect it does, it’s probably the best method outlined here for obtaining sum of two cubes representations.

 

I think what’s going on in all these methods is explained well in [3], particularly the proof of Theorem 2.1.

 

 

References:

 

[1] Amazon.com: Higher Algebra: a Sequel to Elementary Algebra for Schools: Books: Henry Sinclair Hall; Samuel Ratcliff Knight, particularly problem 30 on p. 82.

 

[2] Amazon.com: Rational Points on Elliptic Curves (Undergraduate Texts in Mathematics): Books: Joseph H. Silverman,John Tate, particularly the method outlined on p. 148 as to how to determine rational integer sum of two cubes representations for a rational integer.

 

[3] http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Broughan/broughan25.pdf, “Characterizing the sum of two cubes”.  The author also characterizes differences of rational integer sum of two cubes of rational integers.  The proof of Theorem 2.1 explains well much of what is going on here.