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 ]

## IMO shortlist 2009-N7

- Tahmid Hasan
**Posts:**665**Joined:**Thu Dec 09, 2010 5:34 pm**Location:**Khulna,Bangladesh.

### IMO shortlist 2009-N7

বড় ভালবাসি তোমায়,মা

### Re: IMO shortlist 2009-N7

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

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

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

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Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

- Tahmid Hasan
**Posts:**665**Joined:**Thu Dec 09, 2010 5:34 pm**Location:**Khulna,Bangladesh.

### Re: IMO shortlist 2009-N7

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

anyways thanks for the hint.

anyways thanks for the hint.

বড় ভালবাসি তোমায়,মা

- Phlembac Adib Hasan
**Posts:**1016**Joined:**Tue Nov 22, 2011 7:49 pm**Location:**127.0.0.1-
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### Re: IMO shortlist 2009-N7

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.

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