# a number proof or disproof question

• Nov 24th 2008, 09:01 AM
sah_mat
a number proof or disproof question
if a>0 then we can write $a^3=b^2-c^2$ $b,c\in\mathbb{Z}$
how can ı prove the truthness this equation
• Nov 24th 2008, 11:37 AM
clic-clac
Don't you see that $\forall a \in \mathbb{N}^{*},\ a^{3}=\left( \frac{a^{2}+a}{2}\right)^{2}-\left( \frac{a^{2}-a}{2} \right)^{2}$ ? :p

Actually, you can write $a^{3}=(b+c)(b-c)$, and assume that $b,c$ are non-negative integers: since $b^{2}-c^{2}=(-b)^{2}-(-c)^{2}$ that doesn't change the result.

So a solution would be:
$b+c=a^{2}\ ,\ b-c=a$
You can solve that system in $\mathbb{Q}$, and find $b=\frac{a^{2}+a}{2} ,\ c=\frac{a^{2}-a}{2}$.

But $a^{2}$ and $a$ have the same parity, so the solutions found belong to $\mathbb{N}$, and give us a proof.