Results 1 to 2 of 2

Thread: a number proof or disproof question

  1. #1
    Junior Member
    Nov 2008

    a number proof or disproof question

    if a>0 then we can write$\displaystyle a^3=b^2-c^2$$\displaystyle b,c\in\mathbb{Z}$
    how can ı prove the truthness this equation
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Nov 2008
    Don't you see that $\displaystyle \forall a \in \mathbb{N}^{*},\ a^{3}=\left( \frac{a^{2}+a}{2}\right)^{2}-\left( \frac{a^{2}-a}{2} \right)^{2}$ ?

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

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

    But $\displaystyle a^{2}$ and $\displaystyle a$ have the same parity, so the solutions found belong to $\displaystyle \mathbb{N}$, and give us a proof.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof or disproof the following
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: Apr 3rd 2011, 01:03 PM
  2. Replies: 1
    Last Post: Mar 1st 2010, 08:24 PM
  3. a real number proof question..
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Nov 28th 2008, 08:12 AM
  4. A question about proof of prime number theorem
    Posted in the Number Theory Forum
    Replies: 6
    Last Post: Sep 16th 2008, 10:14 AM
  5. Replies: 3
    Last Post: Apr 7th 2008, 09:20 AM

Search Tags

/mathhelpforum @mathhelpforum