John Conway and Neil L. White have independently made a similar observation, pointing out that the orders of odd constructible regular polygons can all be found in the sequence A001317, the rows of Fermat's triangle modulo 2 interpreted as binary integers. Apparently nobody has any clue whether there are any Fermat primes other than the classic 3, 5, 17, 257, 65537. A very naive probabilistic argument suggests that this list is exhaustive. If there are no more, then the constructible orders are just the first 31 entries in A001317. On Sun, Feb 2, 2020 at 9:41 PM Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:

On 02/02/2020 18:49, Keith F. Lynch wrote:

...

If a regular polygon with N sides is constructable with compass and straight-edge alone, then N expressed in binary is a palindrome followed by zero or more zeros.

Gauss proved that N has to be a product of distinct primes 2^2^n+1 times some power of 2.

The power of 2 gives you zeros at the end. The other factors give you something palindromic: if some product using n<N is palindromic then multiplying by 2^2^N+1 gives you another palindrome because we're taking two copies of its bits and they are separated by just enough 0s that they can't collide.

-- g

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