Can Every Integer Be Expressed as a Sum of Cubes?

[Noo]

I'm studying pure maths/numb theory for the first time (independantly, and from a shallow, brief-ish book with no given solutions, so i have no one else to ask :P ). I just started a day or two ago. The book leaves Induction till a little later, so this should be proved directly, or maybe by contradiction - and only with basic high-school maths.

Even to my novice eyes this problem seems very simple, not simple enough for me, yet, though.

\forall n \in Z, \exists a,b,...,h \in Z such that n = a^{3}+b^{3}+. . .+h^{3}

I must either prove the above is true, or prove that its negation is true.

I'm not sure where to start, and haven't been for 1 or 2 other problems (such as proving 'a^3 -a' is always divisible by 6). I have been thinking of something along the lines of;

Showing \sqrt[3]{a^{3}+b^{3}+c^{3}+d^{3}+e^{3}+f^{3}+g^{3}-n} can't always equate to an integer. But that is random and useless. I'm noy even sure whether the original statement is true or not.

What should i be looking for in trying to prove these things? Or is it mainly that i am not familiar enough with properties of numbers yet? In any case, suggestions/hints/advice or even solutions will all help me in starting out.

 

[HallsofIvy]

First, what exactly is the problem? "Such that" is usually a condition, not a conclusion. Are you trying to prove "If n is an integer, then there exist integers a,b, c..., h that n= a³+ b³+ ... h³" And a, b, ... can be any number of integers, not specifically 8? That probably is NOT the case since then it would be trivial- any number can be written as a sum of "1"s.

 

[Noo]

Yes, Halls, its as you have assumed (or maybe this isn't what you assumed). "For all integers n, there exists 8 integers cubed which sum to n" (That includes 0's). So, it is to prove that statement true, or to prove its negation ("there exists an integer n, for which 8 integers cubed do not sum to n").

 

[Dick]

Yes, Halls, its as you have assumed (or maybe this isn't what you assumed). "For all integers n, there exists 8 integers cubed which sum to n" (That includes 0's). So, it is to prove that statement true, or to prove its negation ("there exists an integer n, for which 8 integers cubed do not sum to n").

Proving something like that would be extremely difficult. You'd better pin your hopes on finding a fairly easy counterexample. There is one. Start expressing small numbers as the sum of cubes and you should find it pretty quickly.

 

[Tedjn]

I could have missed something, but I don't think it's so easy. All integers between -100 and 100 can be represented as the sum of cubes of at most 8 integers, for example.

 

[Dick]

I could have missed something, but I don't think it's so easy. All integers between -100 and 100 can be represented as the sum of cubes of at most 8 integers, for example.

I'll take your word for it, but I think the problem is supposed to be the sum of cubes of nonnegative integers. At least that's the way I took it. Is it Noo? In which case there is an exception.

 

[Tedjn]

Yes, you're right. If the problem is about nonnegative integers, then there is a small exception

 

[Noo]

Proving something like that would be extremely difficult. You'd better pin your hopes on finding a fairly easy counterexample. There is one. Start expressing small numbers as the sum of cubes and you should find it pretty quickly.

I took your advise and have checked all |n| up to 150 so far, i have no counter-example yet. Just to clarify, as i am sure you realize, the sum can include subtractions; so for example the interval [157, 167] is covered by "5^{3} + 4^{3} + (-3)^{3} ^{+}_{-} m, where, since there are 5 remaining components of the sum, each of which can be at 0 or 1, (m=0,1..,5) That is: 125+64-27 = 162, so [(162-5), (162+5)].


Edit: Oh, sorry. I took so long writing this, working LaTex out, that 3 posts arrived before mine. It is for all integers, negative included. And yes, using the interval method i hinted at above i am effectively checking off intervals of the number line. But after i hit 200 I'm giving up hope of finding a counter example:D

P.S. All |n|<205 confirmed.

 

[Dick]

I took your advise and have checked all |n| up to 150 so far, i have no counter-example yet. Just to clarify, as i am sure you realize, the sum can include subtractions; so for example the interval [157, 167] is covered by "5^{3} + 4^{3} + (-3)^{3} ^{+}_{-} m, where, since there are 5 remaining components of the sum, each of which can be at 0 or 1, (m=0,1..,5) That is: 125+64-27 = 162, so [(162-5), (162+5)].


Edit: Oh, sorry. I took so long writing this, working LaTex out, that 3 posts arrived before mine. It is for all integers, negative included. And yes, using the interval method i hinted at above i am effectively checking off intervals of the number line. But after i hit 200 I'm giving up hope of finding a counter example:D

You can give up the search then. If the cubes are allowed to be signed then it is true. In fact you only need 5. I know this because I looked it up, not because I know the proof. I'm pretty sure it's beyond the range of high school algebra. But maybe there is a trick for 8 cubes that makes it accessible. If you are SURE that's the intention, then it's back to the drawing board.

 

[Noo]

Alright, and thanks.

It seems very interesting to me that you only ever need 5. Although in googling just now i learned that all cubes can be expressed as a sum of consecutive odd integers, which i also found very interesting, so perhaps i am easily interested.

 

[Dick]

Alright, and thanks.

It seems very interesting to me that you only ever need 5. Although in googling just now i learned that all cubes can be expressed as a sum of consecutive odd integers, which i also found very interesting, so perhaps i am easily interested.

Since you are so easily interested here's a the '5' proof. I found this in a pdf titled 'Waring's problem, taxicab numbers and other sums of powers'. Google around for it. Start with the identity,

6x=(x+1)^3+(x-1)^3-2*x^3

That proves it for all numbers divisible by six. Now just create some variations on that formula, e.g.

6x+1=(x+1)^3+(x-1)^3-2*x^3+1
6x-1=(x+1)^3+(x-1)^3-2*x^3-1
6x+2=x^3+(x+2)^3-2(x+1)^3-2^3
6x-2=x^3+(x-2)^3-2(x-1)^3+2^3
6x+3=(x-3)^3+(x-5)^3-2(x-4)^3+3^3

There may be some typos in there. Try to find the paper, it might have other interesting stuff in it. I can't find a author on the paper.