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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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 quasi Posts: 12,067 Registered: 7/15/05
Re: rational n-gon inscribed in a unit circle
Posted: Dec 19, 2013 12:56 AM

Richard Tobin wrote:
>quasi wrote:
>>quasi wrote:
>>>
>>> [Revised Conjecture]:
>>>
>>>There does not exist a PRUC pentagon.

>>
>>I'm now pretty sure the revised conjecture will also fail,

I'm no longer so sure.

>>and while I don't have a counterexample,

I still don't.

>I do have a plan of action that has a reasonable chance
>of finding such a counterexample.

The plan seemed promising before I actually tried it.

>>When I get a chance, unless someone else finds a counterexample
>>first, I'll try to implement the plan.

I implemented the plan, but contrary to my prior expectations,
it failed to produce counterexample.

>Have you got anywhere with this?

No.

>I have searched up to radius 500 and found none. All the
>rational pentagons I found

Of course, in this context, "rational pentagons", means
"rational pentagons which can be inscribed in a circle with
a rational radius" -- that is, "rational-unit-cyclic
pentagons" (RUC pentagons), scaled by a rational factor.

>have a rational diagonal, and almost all have a diameter as a
>diagonal. A few have *all* the diagonals rational, so you
>could inscribe a rational pentagram in a rational pentagon,
>e.g. 65: 32 50 66 78 126.

I've found similar examples.

At this point, my level of belief in the truth of the
conjecture has improved to 50-50.

In any case, the conjecture is still open.

quasi

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