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
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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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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.
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- Phlembac Adib Hasan
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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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