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.
-Thanks
Well in that case the smallest cubic number you can use is .
Add 8 of these smallest possible cubes, you get .
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.
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
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.
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?