# Re: OFFTOPIC: Fermat numbers (was Re: Numerical language ...)

From: | Didier Willis <dwillis@...> |

Date: | Thursday, November 19, 1998, 12:24 |

Carlos Thompson wrote:
>
> Didier Willis wrote:
>
> > Beside 2, constructible numbers (that is divisions of a circle
> > that may be constructed with a pair of compasses and a ruler) are
> > prime Fermat numbers and their multiples (The theorem was proven
> > by the mathematician Gauss -- There many other interesting things
> > about Fermat numbers, but it would be really too off-topic here.
> > I may just add that very few Fermat numbers are prime, though
> > the first ones all appear to be prime).
>
> Actually there are infinity of prime Fermat numbers, but density is > very low.

Yes, that's what I wanted to mean. Thanks for your mathematical
precision :*)
> > Fermat numbers are of the form (2^2^a + 1), e.g. 3, 5, 17, 257...
> > Incidentally, 17 (2^2^2 + 1) is constructible, so there is a
> > method (though quite complex, and presumably really too complex
> > for Old Babilonians:) to divise a circle in 17 exact parts with
> > a pair of compasses and a ruler.
>
> I've somewhere read that in the safe box of some University in USA
> is stored the method for constructing a regular 257 sided poligon
> with compasses and ruler.

A method has also been attested for 65537, though nobody never
checked it again (it is *very* long and requires a lot of precision,
and to tell the truth it doesn't have much interest;).
I am presumably wrong in my above assumption, 17 is not so complex
that Babilonians or Greeks could not conceivably construct it
(though I do not known if they actually did it).
Back to conlanging:
Several conlangs I have seen on the web -- including my own
awkward attemps -- have fairly well developed mathematics. In
particular they know the 'infinity' concepts (infinitesimals,
aleph-0...) and I have even seen at least one example of complex
numbers and numerical bases.
However, these concepts are rather late in the history of
mathematics. In comparison, many concepts were discovered
much earlier (pi, algebric numbers...), but paradoxically
they are seldom illustrated in these
conlangs...
Didier.
-- -