65537-gon

[John /Nancy O'Brien]

The following is quoted from "Famous Problems of Geometry and How to
Solve Them" by
Benjamin Bold (Dover Publications :1982):

There is still one unsolved question- for which values of m is
2^(2^m)+1 a prime number?
We know that Fermat's number is prime for m=0,1,2,3,4 and in fact we
have constructed regular
polygons of 3, 5, and 17 sides. For m=3, n=257; and for m=4, n=65,537,
the analysis has also
been accomplished. L.E. Dickson , in a discussion of "Constructions with
Ruler and Compasses",
which appears in "Monographs on Topics of Modern mathematics", states "
The regular 257-gon
has been discussed at great length by Richelot in Crelle's "Journal für
Mathematik", 1832; and
geometrically by Affolter and Pascal in "Rediciniti della R. Accademia
di Napoli ",1887.

The regular polygon of 2^16+1=65,537 sifes has been discussed by
Hermes;"Göttingen
Nachridten,1894"

Also in the April, 1961 issue of "Scientific American", Martin
Gardner describes some of
the topics discussed by H.M.S. Coxeter in a recently published book "An
Introduction to
Geometry". Martin Gardner quoted Professor Coxeter to the effect that
there is at the University
of Göttingen, a large box containing a manuscript showing how to
construct a polygon of 65,537
sides. Gardner also writes "a polygon with a prime number of sides can
be constructed in the
classical manner only if the number is a special type of prime called a
Fermat prime; a prime that
can be expressed as 2^(2^n)+1. Only five such primes are known-
3,5,17,257,65537. The poor
fellow who succeeded in constructing the 65537-gon, Coxeter tells us,
spent ten years at the
task."

End of quotation.

At the time when I read this, I wondered whether anyone actually
checked the manuscript
referred to above to see if the construction of the 65537-gon had
actually been accomplished.
--
John O'Brien

Please remove "nospam." from address if replying by e-mail.

[Steven Anderson]

On Mon, 17 Aug 1998 John /Nancy O'Brien <jnob...@erols.com> wrote:
> Martin Gardner quoted Professor Coxeter to the effect that there

> is at the University of Göttingen a large box containing a manuscript
> showing how to construct a polygon of 65,537 sides... "The poor fellow

> who succeeded in constructing the 65537-gon, Coxeter tells us, spent

> ten years at the task..." At the time when I read this, I wondered

> whether anyone actually checked the manuscript referred to above to
> see if the construction of the 65537-gon hadactually been accomplished.

Here is E.T.Bell's characteristically tendentious version of this old
story (from "Mathematics, Queen and Servant of Science"):

"Simple Euclidean constructions for regular polygons of 17 and 257
sides are available, and an industrious algebraist expended the
better part of his years and a mass of paper in attempting to
construct the regular polygon of 65537 sides. The unfinished
outcome of all this grueling labor was piously deposited in the
library of a German university. Could misguided zeal go further?
...Nobody has yet been so pertinaciously stupid as actually to
carry out the straightedge-and-compass construction for 65537."

[Jeppe Stig Nielsen]

John /Nancy O'Brien wrote:
>[...]

> sides. Gardner also writes "a polygon with a prime number of sides can
> be constructed in the
> classical manner only if the number is a special type of prime called a
> Fermat prime; a prime that
> can be expressed as 2^(2^n)+1. Only five such primes are known-
> 3,5,17,257,65537. The poor

>[...]

Yes.
And if n is a composit number, the n-gon can be constructed if and only
if n is of the form

n = 2^k * p_1 * ... * p_j

where the p's are *different* Fermat primes (known or yet undiscovered).

The first unconstructable polygons are the 7-gon (7 is not Fermat) and the
9-gon (since 9=3*3 and theese Fermat primes aren't different).

--
Jeppe Stig Nielsen, <URL:http://www.imf.au.dk/~jeppesn/>.
My two e-mail addresses, jep...@imf.au.dk and jep...@mi.aau.dk, are
equivalent.