Page **1** of **1**

### IMO shortlist 2009-N7

Posted: **Thu Mar 08, 2012 12:00 am**

by **Tahmid Hasan**

Let $a$ and $b$ be distinct integers greater than $1$. Prove that there exists a positive integer $n$ such that $(a^n−1)(b^n−1)$ is not a perfect square.

[still can't solve

]

### Re: IMO shortlist 2009-N7

Posted: **Thu Mar 08, 2012 8:43 am**

by ***Mahi***

I think Zsigmondi's Theorem can help in this (Maybe).

http://en.wikipedia.org/wiki/Zsigmondy's_theorem

### Re: IMO shortlist 2009-N7

Posted: **Thu Mar 08, 2012 6:49 pm**

by **Tahmid Hasan**

i am trying with LTE and infinite descense and i think i am close to a solution.

anyways thanks for the hint.

### Re: IMO shortlist 2009-N7

Posted: **Fri Mar 09, 2012 2:03 pm**

by **Phlembac Adib Hasan**

I've made a partial solve:if $(a-1)(b-1)$ is not a perfect square, then there are infinite such $n$s.But there is still an unsolved case.I've a question.How will you use infinite descent here?Eager to know.