[SOLVED] Difference of cubes

**courteous** · September 13th 2008, 04:05 AM

$x^3+y^3=208$
x and y are natural numbers.

**Moo** · September 13th 2008, 04:10 AM

Hello,

Originally Posted by courteous

$x^3+y^3=208$
x and y are natural numbers.

That's the sum of cubes

Well, well... The formula is $x^3+y^3=(x+y)(x^2-xy+y^2)$
If you cant remember it, start from $(x+y)^3$ and develop it

Solving for it... ? Not impossible.
Note that $208=2^4*13$

Since x and y are natural numbers, $x+y>0$ and $x^2-xy+y^2>0$ are integers.

So the two factors will be a combination of the divisors of 208. For example 26 and 8 or 13 and 16 or...

-----------------------------------------------
Try all the possible values for x+y among 1,2,4,8,13,16,26,52,104,208 (

)
Then note that $x^2-xy+y^2=(x+y)^2-3xy$ and is respectively among 208,104,...,2,1

Note that $x+y \neq 1,2,4$ otherwise the product can't be large enough (check it

)

If you need more explanations, do tell me. But as far as I can see, it will be a long way to do >.>

**courteous** · September 13th 2008, 06:10 AM

I've correctly titled the thread

, but have mistakenly put + (plus) instead of - (minus)

.

The correct(ed) version:

Originally Posted by courteous

$x^3-y^3=208$
x and y are natural numbers.

The solution is $6^3-2^3=208$ . But how do you get it?
-----------------------------------------------
Moo, about developing from $(x+y)^3$ : what do you do with $3x^2y$ and $3xy^2$ ?

**Moo** · September 13th 2008, 06:26 AM

Moo, about developing from $(x+y)^3$ : what do you do with $3x^2y$ and $3xy^2$ ?

Factor $3xy$ and see what's left

Originally Posted by courteous

I've correctly titled the thread

, but have mistakenly put + (plus) instead of - (minus)

.

The correct(ed) version:

The solution is $6^3-2^3=208$ . But how do you get it?
-----------------------------------------------

Develop (x-y)^3 to get $x^3-y^3=(x-y)(x^2+xy+y^2)$

So we know that x and y are positive -----> $x^2+xy+y^2 > 0$ and $x-y>0$

$x^2+xy+y^2=(x-y)^2+3xy$
Obviously, $x^2+xy+y^2>x-y$

$\begin{array}{c|cc} x-y & x^2+xy+y^2 \\ \hline 1&208& \\ 2&104& \\ 4&52& \\ 8&26& \\ 13&16& \\ 16&13& \rightarrow \text{hey ! } x-y> x^2+xy+y^2 \text{ ? Impossible !} \end{array}$

So you have 5 possibilities to test out (x-y stopping at 13).

For example. If $x-y=1$ then $x^2+xy+y^2=(x-y)^2+3xy=1^2+3xy$
Are there x and y such that $1+3xy=208$ and $x-y=1$ ? That's a linear system of equations

Exactly the same reasoning for the other possibilities !

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[SOLVED] Difference of cubes

Corrected - it is a minus...

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