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Jamie's Mathematical Problems - Vol. 1

Post Icon Posted: Submitted by JebJoya on 4 December 2006 - 2:35pm.

Joined: 2006-10-09
Posts: 327

Consider the set $ P_n=\left\{ \dbinom{x}{n}:0\leq x\leq n\right\} $ (i.e. the $ n $th row of Pascal's Trangle). Prove that $ \forall n,\:\forall x,y\in P_n,\:\Rightarrow (x,y)>1 $.

Thought this might be interesting if anyone fancies doing some random maths?

Jamie

‹ Questions on module/exam failure scenario 1st year geometry and motion ›
Post Icon Posted: 5 December 2006 - 2:13am

Joined: 2006-11-02
Posts: 1090

Number theory, sweet :p

I'm not sure about several details here though: do you mean $ \binom{n}{x} $ in the definition (otherwise it isn't the nth row of Pascal's triangle)? Also $ \gcd(\binom{2}{0},0) = 1 $ or, for instance, $ \gcd(\binom{6}{4},4) = 1 $. Was the problem proving that $ \gcd(\binom{n}{k},k) > 1 $ for all $ n $ and $ k $ or have I misunderstood it?
Post Icon Posted: 5 December 2006 - 4:32pm

Joined: 2006-10-09
Posts: 327

yes, i probably mean $ \binom{n}{x} $ (whatever, doesn't matter), basically that any two numbers on a row of pascal's triangle have hcf>1 (as long as we're not looking at the end values (the 1's)).

Example:

1
1,1
1,2,1
1,3,3,1
1,4,6,4,1
1,5,10,10,5,1
1,6,15,20,15,6,1
...

so, examples, hcf(6,15)=3, hcf(6,20)=2, hcf(15,20)=5... Show it's the same for all rows of pascal's triangle.

Jamie :)
Post Icon Posted: 6 December 2006 - 11:39am

Joined: 2006-10-10
Posts: 520

Awesome! I've got out Knuth's Concrete Mathematics so I'm sure to get a lot of help from that. Just thinking, if anyone does get the answer, mark it with a spoiler tag :)
Post Icon Posted: 21 January 2007 - 3:32pm

Joined: 2006-10-10
Posts: 520

I'm actually going to do this, this weekend ¬¬

Edit: Or not, hehe... Forgot we had G&M for monday...
Post Icon Posted: 21 January 2007 - 4:08pm

Joined: 2006-11-02
Posts: 1090

Oh yeah I forgot about this, I'm going to give it another shot one of these days too.
Post Icon Posted: 8 June 2008 - 3:00pm

Joined: 2007-10-17
Posts: 109

I dare someone to prove this by induction.
Post Icon Posted: 8 June 2008 - 4:09pm

Joined: 2006-10-09
Posts: 327

Necromancy ftw.
Post Icon Posted: 8 June 2008 - 8:50pm

Joined: 2007-10-17
Posts: 109

Oh, is that what thread necromancy means? Sorry about that.

I was perusing old posts and I noticed that no one managed to answer this one. I had a go myself, but got stuck. I'm not good at this kind of problem. I was thinking I'd remind people of this one, although doing that during exam time is probably as much a lapse of judgement as reading old threads in the first place. Since at one point I even wondered whether an inductive argument might be possible, I gave the above cheeky reply.

By the way, this isn't the first time I've comitted necromancy. I resurrected the wikipedia game thread after all. Though perhaps I was let off that as it was my first ever post. Necromancy does seem to happen on the WMS forums sometimes though. As I pointed out earlier today, the cool terms in maths thread is one of the undead.

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