power and factorial

For students of class 11-12 (age 16+)
mutasimmim
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power and factorial

Unread post by mutasimmim » Wed Oct 15, 2014 4:26 pm

Find all pairs of positive integers $ (n, k)$ such that $ n!=(n+1)^k-1 $.

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Phlembac Adib Hasan
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Re: power and factorial

Unread post by Phlembac Adib Hasan » Wed Oct 15, 2014 6:11 pm

Moved to secondary. Too easy for olympiad level.(IMO)
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mutasimmim
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Re: power and factorial

Unread post by mutasimmim » Wed Oct 15, 2014 8:46 pm

You sure?

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Phlembac Adib Hasan
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Re: power and factorial

Unread post by Phlembac Adib Hasan » Thu Oct 16, 2014 9:26 am

Sketch:
From reverse of Wilson's theorem, $n+1=p$ must be prime. Also note that $k\leq p-1$. Suppose $q|p-1$ and is a odd prime. (such a prime exists for $p>3$) $v_q((p-1)!)=\sum\left\lfloor \frac {p-1}{q^i}\right\rfloor$ which will be far greater than $v_q(p^k-1)$ for a large $p$.
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*Mahi*
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Re: power and factorial

Unread post by *Mahi* » Fri Oct 17, 2014 8:37 am

Are you sure it is secondary level? :P This is BdMO forum, not any IMO specific forum; so I think it is better if you move it to at least Higher Secondary level or Olympiad Level Number Theory.
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Phlembac Adib Hasan
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Re: power and factorial

Unread post by Phlembac Adib Hasan » Fri Oct 17, 2014 10:07 am

oka, moved.
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mutasimmim
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Re: power and factorial

Unread post by mutasimmim » Fri Oct 17, 2014 3:58 pm

Ok but how could it possibly be algebra, Mahi vai ?

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*Mahi*
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Re: power and factorial

Unread post by *Mahi* » Fri Oct 17, 2014 7:30 pm

Sorry, it was a typo. I meant to say NT, corrected now.
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