### (2n choose n) is never a perfect power

**Source:**Cute problem sent by Sudeep Kanath

**Problem:**Prove that (2n choose n) is never a perfect power

Update ( 21 June 2014 ):

**Solution:**Posted by Sandeep, Dinesh Krithivasan and Vishal Khatri in comments.

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Update ( 21 June 2014 ):

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This is not a puzzle. So, for those of you who follow this puzzle blog, please bear with me for just one post. Interesting Math in this article though :P Most of my friends already read an article that I wrote more than an year back - " Speak Up " Here, inspired by the movie, The Beautiful Mind, I give a mathematical analysis of asking a girl out. Nice time it is. Feb 10. No plans for Feb 14 and I am sure this article makes me look even more geekier and all the more reason for me to believe that I will be alone, yet again. But what the hell, lets do it! Note: This is not an independent analysis. There are many "mathematics sites" which does "similar" analysis. @Consultants, correct me if I am wrong in my estimates. :P Why is there a need to be selective? From the age of 15, I guess there are approximately 3,600 girls I have liked (On average days, I don't see new girls. But going outside, I like about 30 girls. Saying that I go out once

Once again.. Another coin puzzle after this and this (I wonder whether puzzles have anything else other than hats, kings and coins :D) I heard this from Rushabh Sheth (Mech Sophie) who got it from Vivek Jha (Elec Thirdie). There are 100 coins on the table out of which 50 are tail-face up and 50 are head face up. You are blind folded and there is no way to determine which side is up by rubbing, etc. You have to divide the 100 coins in two equal halves such that both have equal number of coins with tails face up. (This obviously implies that the two have equal number of coins with heads face up) Second part: There are 100 coins on the table out of which 10 are tail-face up and 90 are head face up. You are blind folded and there is no way to determine which side is up by rubbing, etc. You have to divide the 100 coins in two halves (not necessarily equal) such that both have equal number of coins with tails face up. Update (02/01/10): Sorry for inducing confusion in the system. :

Let's say A keep tossing a fair coin, until he get 2 consecutive heads, define X to be the number of tosses for this process; B keep tossing another fair coin, until he get 3 consecutive heads, define Y to be the number of the tosses for this process. 1) Calculate P{X>Y} 2) What's the expected value of X 3) What's the expected value of Y This is probably the hardest puzzle I have ever put on my blog. Hence, I will post its solution in the post directly rather than on comment. Solution: 1) (Solved by me finally after 13 months :)) Make a state diagram. Let the state be (a,b) where a is the number of consecutive heads streak "A" is on currently and b is the number of consecutive heads streak "B" is on currently. So, (0,3) (1,3) are final accepted states and (2,0) (2,1) (2,2) (2,3) are failure states. If you get tails, your contribution to the state reaches to "0" f(State) = P(X>Y | "State" configuration initially)

There is always atleast one prime between n and 2n. (Bertrand's postulate). These primes occur only once in factorization of 2nCn. So, 2nCn can never be a perfect power.

ReplyDeleteThanks

DeleteDirect consequence of Bertrand's postulate. After canceling out one of the n! in the denominator, the numerator will be the product of (n+1)(n+2)... up to 2n. By Bertrand's postulate, there is a prime in this group of numbers, say p. Then, (2n choose n) is divisible by p but not any powers of p and so cannot be a perfect power.

ReplyDeleteThis is probably nuking a mosquito though - there ought to be a simple proof.

Most proofs of Bertrand's postulate (at least the ones on wiki) start by studying the prime factorization of 2nCn. So, this reasoning is likely to be circular.

DeleteThanks.

DeleteIt will always be some power of 2 multiplied with some odd number. So, it can't be a perfect power.

ReplyDelete?? Wrong solution!

DeleteJust saw your comment on this..I agree, I have written it wrong... just have to work out again to what i wanted to write. Thanks for bringing it to notice :-)

DeleteI meant to say that it will always be a power of 2 multiplied with some odd numbers of which at least one would a prime. And after reading the comments above and below, I noticed that the prime would be lying in n to 2n.

DeleteSo, I wrote it wrong, Thank you for pointing out :-)

There is always a prime between n and 2n. Hence!

ReplyDeleteThanks

DeleteBertrand's postulate it is !!!

ReplyDeletehttp://en.wikipedia.org/wiki/Bertrand%27s_postulate