a number proof or disproof question

**sah_mat** · November 24th 2008, 08:01 AM

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

**clic-clac** · November 24th 2008, 10:37 AM

Don't you see that $\forall a \in \mathbb{N}^{*},\ a^{3}=\left( \frac{a^{2}+a}{2}\right)^{2}-\left( \frac{a^{2}-a}{2} \right)^{2}$ ?

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.

Thread: a number proof or disproof question

LinkBack

Thread Tools

Display

a number proof or disproof question

Similar Math Help Forum Discussions

Proof or disproof the following

Contradiction, counterexample, when and how to use each form of proof and disproof?

a real number proof question..

A question about proof of prime number theorem

proof that 26 is the only number between a cubed and a squared number

Search Tags