# I need some help with a discrete math proof

• Sep 27th 2008, 01:47 PM
NidhiS
I need some help with a discrete math proof
um well the question goes like this.

Professor X thinks that all natural numbers can be written as a sum of 8(not necessarily distinct) cubes.Prove that he's wrong. and well you can use something like -1 because it is not in the domain of natural numbers.I need some help here.

(Worried)

-Thanks
• Sep 27th 2008, 08:38 PM
Prove It
Quote:

Originally Posted by NidhiS
um well the question goes like this.

Professor X thinks that all natural numbers can be written as a sum of 8(not necessarily distinct) cubes.Prove that he's wrong. and well you can use something like -1 because it is not in the domain of natural numbers.I need some help here.

(Worried)

-Thanks

Can the cubes be any real number, or do they have to be natural numbers as well?
• Sep 27th 2008, 08:42 PM
NidhiS
well the cubes are all natural numbers as well such as 1,8,27 etc
• Sep 27th 2008, 08:45 PM
Prove It
Quote:

Originally Posted by NidhiS
well the cubes are all natural numbers as well such as 1,8,27 etc

Well in that case the smallest cubic number you can use is $\displaystyle 1 = 1^3$.

Add 8 of these smallest possible cubes, you get $\displaystyle 8 \times 1^3 = 8$.

So the smallest possible number you can get by adding 8 cubes is 8.

Can you think of a way to get 1, 2, 3, 4, 5, 6 or 7?

Obviously there isn't one.

Counter-example has been provided, Professor X has been proven wrong.
• Sep 27th 2008, 08:51 PM
NidhiS
well 0 is a natural number as well.
so 7 can be written as 7 times 1's cube and one times 0's cube.It's not easy to prove this with the method o exhaustion.
• Sep 27th 2008, 08:59 PM
Prove It
Quote:

Originally Posted by NidhiS
well 0 is a natural number as well.
so 7 can be written as 7 times 1's cube and one times 0's cube.It's not easy to prove this with the method o exhaustion.

Not necessarily.

The set of natural numbers does not always include 0. It depends on how a person defines it.

Natural number - Wikipedia, the free encyclopedia
• Sep 27th 2008, 09:05 PM
Jhevon
Quote:

Originally Posted by Prove It
Not necessarily.

The set of natural numbers does not always include 0. It depends on how a person defines it.

Natural number - Wikipedia, the free encyclopedia

indeed. but the problem is obviously trivial if we say zero is not included. so i guess we should just assume it is
• Sep 27th 2008, 09:07 PM
Prove It
Quote:

Originally Posted by Jhevon
indeed. but the problem is obviously trivial if we say zero is not included. so i guess we should just assume it is

Yes, but if you include 0 then NidhiS showed you CAN get any natural number by the sum of 8 cubes...

The answer must lie with whether or not 0 is included...
• Sep 27th 2008, 09:11 PM
NidhiS
um

well it says
"or an element of the set {0, 1, 2, 3, ...} (the non-negative integers). The former is generally used in number theory, while the latter is preferred in mathematical logic, set theory, and computer science. A more formal definition will follow."

coutesy wikipedia and that's how my prof explained the question.He said zero's included tho.
• Sep 27th 2008, 09:14 PM
NidhiS
I think we have to prove it by this method called "Method of generalizing using generic particular". wherein we take an arbitrary digit adn prove that the result evaluates to alse or this number.

Lets say we take a number called x

8 times x's cube = m

so x= (m/8)^1/3

does it prove anything? I mean is the RHS value for this expression not a natural number?
• Sep 27th 2008, 09:31 PM
NidhiS

There has to be somebody whose atleast as smart as me and can help me on this question.XD

(Makeup)
• Sep 27th 2008, 11:51 PM
CaptainBlack
Quote:

Originally Posted by NidhiS
um well the question goes like this.

Professor X thinks that all natural numbers can be written as a sum of 8(not necessarily distinct) cubes.Prove that he's wrong. and well you can use something like -1 because it is not in the domain of natural numbers.I need some help here.

(Worried)

-Thanks

Consider 23, which is the smallest exception to Landau's Theorem on the representation of natural numbers as sum of eight cubes.

RonL
• Sep 28th 2008, 12:00 AM
bkarpuz
Quote:

Originally Posted by NidhiS
um well the question goes like this.

Professor X thinks that all natural numbers can be written as a sum of 8(not necessarily distinct) cubes.Prove that he's wrong. and well you can use something like -1 because it is not in the domain of natural numbers.I need some help here.

(Worried)

-Thanks

Professor X does not claim that negative integers can not be written as a sum of $\displaystyle 8$ cubes.
So why do you think of using $\displaystyle -1$?
• Sep 28th 2008, 06:49 AM
NidhiS
Well negative integer does not fall in the domain of Natural numbers.Well you cant use negative integers.