quasi
Posts:
11,066
Registered:
7/15/05


Re: rational ngon 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, "rationalunitcyclic 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 5050.
In any case, the conjecture is still open.
quasi

