IMO Shortlist 2009 problem N7
Dodao/la:
arhiva2. travnja 2012. Let
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
be distinct integers greater than
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
. Prove that there exists a positive integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
such that
![\left(a^n-1\right)\left(b^n-1\right)](/media/m/5/d/0/5d0642845fff40e27352c95318c3238c.png)
is not a perfect square.
Proposed by Mongolia
%V0
Let $a$ and $b$ be distinct integers greater than $1$. Prove that there exists a positive integer $n$ such that $\left(a^n-1\right)\left(b^n-1\right)$ is not a perfect square.
Proposed by Mongolia
Izvor: Međunarodna matematička olimpijada, shortlist 2009