Comments for The Math Less Traveled http://mathlesstraveled.com Explorations in mathematical beauty Fri, 18 May 2012 19:30:04 +0000 hourly 1 http://wordpress.com/ Comment on Carnival of Mathematics 86 by A math carnival here?? « Random Walks http://mathlesstraveled.com/2012/05/08/carnival-of-mathematics-86/#comment-10169 Fri, 18 May 2012 19:30:04 +0000 http://mathlesstraveled.com/?p=1589#comment-10169 [...] if you haven’t done so yet, please go check out the current carnival at the Math Less [...]

]]> Comment on Fibonacci multiples by Brent http://mathlesstraveled.com/2012/05/15/fibonacci-multiples/#comment-10140 Wed, 16 May 2012 22:54:10 +0000 http://mathlesstraveled.com/?p=1639#comment-10140 Cool proof!

I really wish there were a “preview comment” button too. But wordpress.com does not support it. =(

]]> Comment on Fibonacci multiples by JRH http://mathlesstraveled.com/2012/05/15/fibonacci-multiples/#comment-10139 Wed, 16 May 2012 22:46:09 +0000 http://mathlesstraveled.com/?p=1639#comment-10139 For M = \left[\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array}\right], M^{n} = \left[\begin{array}{ll} F_{n+1} & F_{n} \\ F_{n} & F_{n-1} \end{array}\right]. (Proof by induction left as an exercise for the reader.)

Now assume that F_{km} = K_{k}F_{m} for k \geq 1, which can be proven for the base cases of 1 and 2 with K_{1} = 1 and K_{2} = [F_{m+1}+F_{m-1}] by expanding M^{2m} = M^{m}M^{m}.

Then M^{(k+1)m} = M^{km}M^{m} and so F_{(k+1)m} = F_{km+1}F_{m} + F_{km}F_{m-1} = K_{k+1}F_{m}, with K_{k+1} = [F_{km+1} + K_{k}F_{m-1}].

I really wish there were a “preview comment” button so I could double check all of that LaTeX before posting.

]]> Comment on Fibonacci multiples by Brent http://mathlesstraveled.com/2012/05/15/fibonacci-multiples/#comment-10133 Wed, 16 May 2012 13:56:22 +0000 http://mathlesstraveled.com/?p=1639#comment-10133 Yes, I can see how that would work. It’s different from both of my proofs (one of which can also be formulated in terms of those pictures, now that I have thought about it more carefully).

Now, how about that proof using the closed form of fibonacci and symmetric polynomials? ;-)

]]> Comment on Fibonacci multiples by Matt Gardner Spencer http://mathlesstraveled.com/2012/05/15/fibonacci-multiples/#comment-10132 Wed, 16 May 2012 13:48:17 +0000 http://mathlesstraveled.com/?p=1639#comment-10132 The key to my proof is using pictures like those here: http://mathlesstraveled.com/2011/05/24/post-without-words-1/ (especially those posted by Xander in the comments) to prove
F_{mn}=F_n\cdot F_{(m-1)n+1}+F_{n-1}\cdot F_{(m-1)n} and then using induction.

]]> Comment on Fibonacci multiples by Joseph Nebus http://mathlesstraveled.com/2012/05/15/fibonacci-multiples/#comment-10128 Wed, 16 May 2012 01:48:54 +0000 http://mathlesstraveled.com/?p=1639#comment-10128 Reblogged this on nebusresearch and commented:
I have not, as far as I remember, encountered this theorem before. And for the time I’ve had to think about it I realize I’ve got no idea how to prove it. However, it’s a neat little result that makes me smile to hear about, and theorems that bring smiles are certainly worth sharing.

]]> Comment on Fibonacci multiples by Joseph Nebus http://mathlesstraveled.com/2012/05/15/fibonacci-multiples/#comment-10127 Wed, 16 May 2012 01:46:25 +0000 http://mathlesstraveled.com/?p=1639#comment-10127 I’d never heard of this theorem before and I’m captivated by it. I’d have expected to run across it in those occasional lists of cool mathematics trivia.

]]> Comment on Fibonacci multiples by Brent http://mathlesstraveled.com/2012/05/15/fibonacci-multiples/#comment-10125 Tue, 15 May 2012 23:11:48 +0000 http://mathlesstraveled.com/?p=1639#comment-10125 Nope, but if you know of such a proof I’d love to hear it! One of the proofs definitely uses induction, and can probably be formulated in terms of dominos, though I haven’t thought about the details. The other one is more elementary.

I would also like to see a proof involving monimos, dominos, and geronimos.

]]> Comment on Fibonacci multiples by Matt Gardner Spencer http://mathlesstraveled.com/2012/05/15/fibonacci-multiples/#comment-10122 Tue, 15 May 2012 21:28:52 +0000 http://mathlesstraveled.com/?p=1639#comment-10122 Sounds exciting! I’m guessing that neither of your proofs uses the closed form of fibonacci and symmetric polynomials. But how about induction and monimos and dominos?

]]> Comment on Fibonacci multiples by Dennis http://mathlesstraveled.com/2012/05/15/fibonacci-multiples/#comment-10120 Tue, 15 May 2012 16:59:25 +0000 http://mathlesstraveled.com/?p=1639#comment-10120 I wonder…

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