## power and factorial

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

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

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### Re: power and factorial

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

You sure?

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### Re: power and factorial

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

Are you sure it is secondary level? 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.

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

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### Re: power and factorial

oka, moved.
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mutasimmim
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### Re: power and factorial

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

*Mahi*
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### Re: power and factorial

Sorry, it was a typo. I meant to say NT, corrected now.
Use $L^AT_EX$, It makes our work a lot easier!