fom
Posts:
1,968
Registered:
12/4/12


Re: rational ngon 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 ngon rational if all edge lengths are rational. >>>>> For n > 6, does there exist a rational ngon which can be >>>>> inscribed in a unit circle? >>>> >>>> Why not multiply your question by the LCM? When can ngons >>>> whose sides are integers be inscibed inside of a circle of >>>> integer radius? >>> >>> 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.

