Conic-Constructible Regular Polygons

Antreas P. Hatzipolakis xpolakis at
Wed Dec 15 21:20:44 CET 1999

John Conway wrote:

>On Tue, 14 Dec 1999, Antreas P. Hatzipolakis wrote:
>> Does anyone know which is the sequence of the R. polygons which are
>> conic-constructible? (since right now I have not handy the Math. Int.
>> volumes to check)
>   I think it's obvious that the answer is those for which n has the form
>2^a.3^b.p.q.r....   ,  where  p,q,r,...  are distinct primes of the
>form  2^x.3^y + 1.   Yes, that IS obvious.

And I think that what is obvious for JHC is not obvious for everyone else! :-)

There is a theorem (by Vieta) which says:

(conic-constructibility) <=> (marked ruler constructibility: trisect an angle)

Now, for the (marked ruler constructibility) the formula is the one given
by Conway above.

So, the Conics sequence is: 7,9,13,14,18,19,21,26,27,28,35,36,37,38,39,

But this sequence is not found in EIS:
I am sorry, but the terms
do not match anything in the table.

Note that the primes of the form 2^x * 3^y + 1 are called Pierpoint Primes

Note also that the "Greek Sequence" (2^a * 3^b * 5^c; where:
a = 0,1,2,3..., b,c in {0,1}, excluding the terms 1,2; that is:
(a,b,c) =/= (0,0,0), (1,0,0)):

was not found in EIS either.
I am sorry, but the terms
do not match anything in the table.

George E. Martin: Geometric Constructions.
New York etc: Springer, 1997, p. 140


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