So, a maker's mark could not include a regular septagon or nonagon (7 or 9 sides) since it's mathematically provable that neither of those figures could be constructed with only compass and straightedge.

A brief diversion from my mathematical bag of tricks...

Everyone knows that a regular hexagon can be easily constructed in a circle using only a compass and a straightedge. With the compass set to the radius of the circle, simply walk it around the circle striking off points on the circumference. There will be exactly six points. Connect these points with straight lines and the result will be a regular hexagon.

This raises the question of what other regular polygons can be constructed using only compass and straightedge? Obviously, three and four sides are easy and we know from above that six is possible. What about five or seven?

Karl Friederich Gauss, the German mathematical prodigy and genius, solved the generalized problem. He proved that a regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of *distinct* Fermat primes (including none).

A Fermat prime is a Fermat number of the form

Fk = 2^(2^k)+1

that is also a prime. The known Fermat primes are:

k Fk

0 3
1 5
2 17
3 257
4 65537

Not all Fermat numbers are prime. The list above includes all the currently known Fermat primes. For example, for n = 5, we have the Fermat number 4,294,967,297 which has the factors 641 and 6,700,417 and so is not prime.

Also, the restriction to *distinct* primes is important. A 9-sided polygon cannot be constructed. 9 has prime factors 3*3 and, while 3 is a Fermat prime, ALL the factors of 9 must be DISTINCT Fermat primes for it to be constructible.

So, based on the above, a regular septagon cannot be constructed, but a pentagon can.