Home » Ask & Discuss » Mathematics. » Algebra « Back to Discussion

Algebra

Cool goIITian

Joined:	24 Oct 2008
Post:	69

6 Feb 2010 09:37:44 IST

0 People liked this

416

P.T for any integer n>1, n! Is not a perfect square

None

P.T for any integer n>1, n! Is not a perfect square

Share this article on:

Comments (2)

Hari Shankar

Forum Expert

Joined: 28 Feb 2007

Posts: 2173

6 Feb 2010 11:25:24 IST

Like 3 people liked this

From Bertrand-Chebyshev theorem, if k is a natural there exists at least one prime p such that k<p<2k-2

This prime will divide both (2k-1)! and 2k!.

Now we've got to see to what power it appears which is given by the Legendre function.

$ord_p(n!) = \sum_{k=1}^{\infty} \left[\frac{n}{p^k} \right]$

So if n = 2k or 2k-1 then

$1 \le \frac{n}{p} < \frac{2k}{k} =2$

which means ord_p(n!) =1 .

That means these primes appear to the power 1 in the prime factorisation of n!. But for a perfect square, every prime appears to an even power.

Hence n! cannot be of the form a^b with b>1 i.e it cannot be a perfect power of an integer. In particular it cannot be a perfect square.

left the site ..................

Cool goIITian

Joined: 30 Dec 2009

Posts: 88

6 Feb 2010 11:52:09 IST

Like 0 people liked this

moe elaborate

Quick Reply

Reply

	Some HTML allowed.
	Keep your comments above the belt or risk having them deleted.
	Signup for a avatar to have your pictures show up by your comment
	If Members see a thread that violates the Posting Rules, bring it to the attention of the Moderator Team

Free Sign Up!

Preparing for IIT-JEE ?

Arihant Revision Package for IIT JEE - Books, Practice Tests + Rank Predictor

@ INR 1,995/-

For Quick Info

Find Posts by Topics

Physics.

Topics

9390

20087

3671

2186

7614

2826

2636

Mathematics.

Chemistry.

Biology

Institutes

Parents

Board

Fun Zone