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 Post subject: Every Positive IntegerPosted: Tue, 17 Apr 2012 15:27:52 UTC
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Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
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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 Post subject: Re: Every Positive IntegerPosted: Tue, 17 Apr 2012 15:37:43 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6005
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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 and . Alternatively, use greedy.

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 Post subject: Re: Every Positive IntegerPosted: Tue, 17 Apr 2012 15:39:20 UTC
 Senior Member

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
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.

_________________
math puns are the first sine of madness
-JDR

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 Post subject: Re: Every Positive IntegerPosted: Tue, 17 Apr 2012 18:41:21 UTC
 Senior Member

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
Sorry for the double post~

outermeasure wrote:
I suspect you want distinct powers

Yessir

outermeasure wrote:
Spoiler:
Repeated application of and .

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!

_________________
math puns are the first sine of madness
-JDR

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