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rdj5933mile5math64
 Post subject: Every Positive Integer
PostPosted: Tue, 17 Apr 2012 15:27:52 UTC 
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Prove that every positive integer can be written as a sum of finite distinct integral powers of the golden ratio.

The main idea behind my solution:
Spoiler:
Induction

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Last edited by rdj5933mile5math64 on Tue, 17 Apr 2012 15:38:59 UTC, edited 1 time in total.

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outermeasure
 Post subject: Re: Every Positive Integer
PostPosted: Tue, 17 Apr 2012 15:37:43 UTC 
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rdj5933mile5math64 wrote:
Prove that every positive integer can be written as a sum of finite integral powers of the golden ratio.

The main idea behind my solution:
Spoiler:
Induction


Trivially,
Spoiler:
Every positive integer is a sum of 1s.


I suspect you want distinct powers instead, in which case
Spoiler:
Repeated application of 2=\varphi+\varphi^{-2} and 1+\varphi=\varphi^2. Alternatively, use greedy.

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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rdj5933mile5math64
 Post subject: Re: Every Positive Integer
PostPosted: Tue, 17 Apr 2012 15:39:20 UTC 
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outermeasure wrote:
rdj5933mile5math64 wrote:
Prove that every positive integer can be written as a sum of finite integral powers of the golden ratio.

The main idea behind my solution:
Spoiler:
Induction


Trivially,
Spoiler:
Every positive integer is a sum of 1s.

Fixed. :P

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math puns are the first sine of madness
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rdj5933mile5math64
 Post subject: Re: Every Positive Integer
PostPosted: Tue, 17 Apr 2012 18:41:21 UTC 
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Sorry for the double post~

outermeasure wrote:
I suspect you want distinct powers


Yessir

outermeasure wrote:
Spoiler:
Repeated application of 2=\varphi+\varphi^{-2} and 1+\varphi=\varphi^2.


This is what I did! :)

outermeasure wrote:
Spoiler:
Alternatively, use greedy.


That's the monovariant solution! lolol I was looking with this for awhile trying to find something (haven't heard of greedy algroithm before). Thank You! :D

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