power and factorial

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power and factorial
Find all pairs of positive integers $ (n, k)$ such that $ n!=(n+1)^k1 $.
 Phlembac Adib Hasan
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Re: power and factorial
Moved to secondary. Too easy for olympiad level.(IMO)
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 Phlembac Adib Hasan
 Posts: 1016
 Joined: Tue Nov 22, 2011 7:49 pm
 Location: 127.0.0.1
 Contact:
Re: power and factorial
Sketch:
From reverse of Wilson's theorem, $n+1=p$ must be prime. Also note that $k\leq p1$. Suppose $qp1$ and is a odd prime. (such a prime exists for $p>3$) $v_q((p1)!)=\sum\left\lfloor \frac {p1}{q^i}\right\rfloor$ which will be far greater than $v_q(p^k1)$ for a large $p$.
From reverse of Wilson's theorem, $n+1=p$ must be prime. Also note that $k\leq p1$. Suppose $qp1$ and is a odd prime. (such a prime exists for $p>3$) $v_q((p1)!)=\sum\left\lfloor \frac {p1}{q^i}\right\rfloor$ which will be far greater than $v_q(p^k1)$ for a large $p$.
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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.
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Nur Muhammad Shafiullah  Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah  Mahi
 Phlembac Adib Hasan
 Posts: 1016
 Joined: Tue Nov 22, 2011 7:49 pm
 Location: 127.0.0.1
 Contact:

 Posts: 107
 Joined: Sun Dec 12, 2010 10:46 am
Re: power and factorial
Ok but how could it possibly be algebra, Mahi vai ?
Re: power and factorial
Sorry, it was a typo. I meant to say NT, corrected now.
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Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah  Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah  Mahi