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Wikipedia tells me that the 11-gon was found to be neusis constructible in 2014, and the link given doesn't seem to be a crank, but the actual method is behind a paywall. (Interestingly, the page itself states positively that the 41-gon and 61-gon are not neusis constructible, which I would think would follow.) While looking for more details, I stumbled upon these lecture notes that purport to prove that any neusis-constructible length can be found in a Galois extension of $\Bbb Q$ of dimension $2^a3^b$, which would seemingly preclude the edge of an 11-gon. So I was just wondering, who is right? If the lecture notes linked are wrong, where are they wrong? If the lecture notes are right, how did the error in Benjamin and Snyder get to such a high level, and if anyone is familiar with their work, what was it?

user361424
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  • See the article by Baragar referenced by the Benjamin-Snyder paper on the 11-gon. – i. m. soloveichik Dec 02 '16 at 23:59
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    The Wikipedia page has been edited. Among numbers whose Euler totients have a maximum prime factor of 5, those described by 11 times 1, 2, or a "classically" neusis-constructible case are now listed as neusis constructible; but all others are open problems. – Oscar Lanzi Feb 27 '18 at 03:10

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You can find some good slides of one of the author about this here

My understanding is this:

Unmarked ruler - usual theory, only some extensions of degree $2^n$ can be done.

Marked Ruler but **only allowed to use marks on lines **, only some extensions of degree $2^n3^m$ can be done.

But, if one allows the use of the marks between circles, or between a line and a circle, more is possible.

I think they prove that with marks allowed between a lines or between a line and a circle (but not between two circles) they can do a construction if and only if the intermediate degrees are up to degree 6. That means that some extensions of the form $2^n3^m5^k$ are doable.

N. S.
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    With marks between two circles you are still limited to degree 6. Baragar outlines the proof in his 2002 paper (accessible here), but going through the details would cause an explosion of algebra. The limitation blows up any possibility of neusis-constructing regular 23- or 29-gon, but leaves open the 11-gon which Benjamin and Snyder later solved. – Oscar Lanzi Feb 27 '18 at 02:23
  • If I recall, the minimum polynomial for the cosine of the 23-gon angle is of 11th degree. 11 obviously has a prime factor greater than 5. – Michael Ejercito Feb 19 '24 at 19:20
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There is no error in Benjamin and Snyder. I had actually discussed this problem with them, even did my own analysis, and it is correct. See here for a description of what they did.

Nor are the lecture notes wrong. These are perfectly correct within their range of assumptions -- which is verging between two lines. The third paragraph of the lecture notes explicitly identifies this limitation. Benjamin and Snyder, and Baragar before them, allow verging with circles and thus go beyond the range of the lecture notes. When you allow verging with circles, the theories of Baragar and of Benjamin and Snyder allow some quintic and sextic equations to be solved along with the cubic and quar-tic equations enabled by just verging between lines.

Be careful not to over-read what is found in Baragar and in Benjamin and Snyder. We now know only that at least some quintic equations are neusis-solvable. We have no theoretical knowledge covering all quintic equations and, therefore, no general theory for constructing all "quintic" regular polygons. We merely got a hit on one specific case of such polygons.

Oscar Lanzi
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