power and factorial
Posted: 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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power and factorial
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Re: power and factorial
Posted: Wed Oct 15, 2014 6:11 pm
Moved to secondary. Too easy for olympiad level.(IMO)
Re: power and factorial
Posted: Wed Oct 15, 2014 8:46 pm
You sure?
Re: power and factorial
Posted: 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 $p$.
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 $p$.
Re: power and factorial
Posted: Fri Oct 17, 2014 8:37 am
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.
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.Re: power and factorial
Posted: Fri Oct 17, 2014 10:07 am
oka, moved.
Re: power and factorial
Posted: Fri Oct 17, 2014 3:58 pm
Ok but how could it possibly be algebra, Mahi vai ?
Re: power and factorial
Posted: Fri Oct 17, 2014 7:30 pm
Sorry, it was a typo. I meant to say NT, corrected now.