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Topic: rational n-gon inscribed in a unit circle
Replies: 59   Last Post: Dec 19, 2013 12:56 AM

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 fom Posts: 1,968 Registered: 12/4/12
Re: rational n-gon inscribed in a unit circle
Posted: Dec 11, 2013 4:33 PM

On 12/11/2013 3:04 PM, quasi wrote:
> fom wrote:
>> quasi wrote:
>>> scattered wrote:
>>>> quasi wrote:
>>>>>
>>>>> Call an n-gon rational if all edge lengths are rational.
>>>>> For n > 6, does there exist a rational n-gon which can be
>>>>> inscribed in a unit circle?

>>>>
>>>> Why not multiply your question by the LCM? When can n-gons
>>>> whose sides are integers be inscibed inside of a circle of

>>>
>>> Yes, it's the same question.
>>>
>>> I just chose to state the question using radius 1 and rational
>>> edge lengths.

>>
>> If you state the question so that both the radius and the side
>> are even natural numbers, would this not reduce to a question
>> concerning Pythagorean triples?

>
> At some level, surely.
>
> But not in any immediate way.
>
> To attack the problem, rescaling so that all rational lengths
> are integers is a fine first step, but no instantly apparent
> integral right triangles arise directly from such a rescaling.
>
> quasi
>

Well, I had been thinking more in terms of constraints
on possible solutions.

"in order that given a natural number n there
exist a pythagorean triangle with hypotenuse n,
it is necessary and sufficient that the number
n have at least one prime factor of the form
4k + 1"

This is one of the remarks from Sierpinski.

It narrows the search a small bit.

But, you are correct that no instantly apparent
solution would present itself from what I have
about these triples. The statement above would
be about the only generally applicable statement
that pops up.

Date Subject Author
12/10/13 quasi
12/10/13 ross.finlayson@gmail.com
12/10/13 quasi
12/11/13 quasi
12/11/13 quasi
12/11/13 quasi
12/12/13 quasi
12/12/13 Helmut Richter
12/12/13 quasi
12/11/13 scattered
12/11/13 quasi
12/11/13 fom
12/11/13 fom
12/11/13 quasi
12/11/13 fom
12/11/13 Richard Tobin
12/11/13 Richard Tobin
12/12/13 quasi
12/12/13 Richard Tobin
12/12/13 Richard Tobin
12/12/13 quasi
12/12/13 Brian Q. Hutchings
12/13/13 quasi
12/13/13 Brian Q. Hutchings
12/12/13 Thomas Nordhaus
12/12/13 Richard Tobin
12/12/13 quasi
12/12/13 Richard Tobin
12/12/13 quasi
12/12/13 quasi
12/13/13 Richard Tobin
12/13/13 Richard Tobin
12/13/13 quasi
12/13/13 Richard Tobin
12/13/13 quasi
12/13/13 Richard Tobin
12/13/13 quasi
12/13/13 quasi
12/12/13 Richard Tobin
12/13/13 quasi
12/13/13 Richard Tobin
12/13/13 quasi
12/14/13 quasi
12/14/13 quasi
12/14/13 quasi
12/14/13 quasi
12/14/13 Richard Tobin
12/15/13 quasi
12/15/13 quasi
12/15/13 Richard Tobin
12/15/13 David Bernier
12/15/13 quasi
12/18/13 Richard Tobin
12/18/13 ross.finlayson@gmail.com
12/19/13 quasi
12/14/13 Richard Tobin
12/14/13 quasi
12/14/13 Richard Tobin
12/14/13 ross.finlayson@gmail.com
12/15/13 Brian Q. Hutchings