IMO shortlist 2009-N7

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Tahmid Hasan
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IMO shortlist 2009-N7

Unread post by Tahmid Hasan » Thu Mar 08, 2012 12:00 am

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 :x]
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*Mahi*
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Re: IMO shortlist 2009-N7

Unread post by *Mahi* » Thu Mar 08, 2012 8:43 am

I think Zsigmondi's Theorem can help in this (Maybe).
http://en.wikipedia.org/wiki/Zsigmondy's_theorem
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Tahmid Hasan
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Re: IMO shortlist 2009-N7

Unread post by Tahmid Hasan » Thu Mar 08, 2012 6:49 pm

i am trying with LTE and infinite descense and i think i am close to a solution.
anyways thanks for the hint. :)
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Phlembac Adib Hasan
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Re: IMO shortlist 2009-N7

Unread post by Phlembac Adib Hasan » Fri Mar 09, 2012 2:03 pm

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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