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arxiv: 2505.14865 · v3 · submitted 2025-05-20 · 🧮 math.MG

Regular polygons

J. Mainik This is my paper

Pith reviewed 2026-05-22 13:34 UTC · model grok-4.3

classification 🧮 math.MG
keywords regular polygonscompass and straightedgeFermat primes65537-gongeometric constructionGauss-Wantzel theorem
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The pith

The regular 65537-gon can be constructed exactly with compass and straightedge.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows a complete sequence of compass and straightedge operations that produces the regular polygon with exactly 65537 sides. This matters because Gauss proved in theory that such a polygon is constructible when the number of sides is a power of two times distinct Fermat primes, and 65537 is the largest known such prime. Earlier work by Hermes described the construction but remained unpublished due to its length, leaving a practical gap that this new approach closes by detailing every step without omissions.

Core claim

The central claim is that the regular 65537-gon admits an explicit, gap-free construction using only compass and straightedge operations, obtained by a new detailed analysis of the case n=65537 that builds directly on the Gauss-Wantzel characterization of constructible regular polygons.

What carries the argument

The new detailed approach to the construction task for regular n-gons where n is a Fermat prime, applied to produce the full sequence for n=65537.

Load-bearing premise

The sequence of compass and straightedge operations contains no hidden errors or omitted intermediate steps.

What would settle it

Carrying out the full sequence of steps and checking whether the resulting figure has all sides equal and all interior angles equal to the correct value for a regular 65537-gon.

read the original abstract

The construction of regular polygons with a compass and straightedge is a well-known task and this problem has interested mathematicians for a long time. In particular, for a long time they could not answer the question of whether is it possible to construct a regular 17-gon with a compass and straightedge. C. F. Gauss solved this problem in 1796. He proved later that it is possible to construct with a compass and straightedge the regular polygons with $n=2^m n_1\cdots n_l$ sides, where $n_1,\cdots, n_l$ are different prime numbers of the form $\; n_k=2^{2^{\nu_k}}+1$. P. Wantzel proved in 1837 that only these regular polygons can be constructed. Essential is here the construction of the regular polygons with $n_k=2^{2^{\nu_k}}+1$ sides. The currently known prime numbers of the form $n=2^{2^{\nu}}+1$ are $3, 5, 17, 257$ and $65537$. In the paper we present a new approach for solving this task. Among other things we analyze in detail the case of $n=65537$. J. G. Hermes announced in 1894 that he had a full description of the construction of the 65537-gon. This was the result of 10 years of work, but his text was too extensive and was never published. We show exactly and without gaps how the regular 65537-gon can be constructed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper recalls the classical results of Gauss and Wantzel on constructible regular polygons, notes the known Fermat primes, and presents a new approach to the compass-and-straightedge construction problem. It claims to analyze the case n=65537 in detail and to exhibit an exact, gap-free construction sequence for the regular 65537-gon, filling the gap left by Hermes' unpublished work.

Significance. An explicit, verifiable construction for the 65537-gon would be of genuine historical and technical interest, as it would supply the first published complete sequence realizing the full tower of quadratic extensions of degree 2^16. If the new approach is both correct and more transparent than prior unpublished attempts, it could serve as a useful reference for explicit constructions in cyclotomic fields.

major comments (1)
  1. The central claim that the construction of the regular 65537-gon is shown 'exactly and without gaps' (abstract and the dedicated analysis section) is not supported by any enumerated sequence of compass-and-straightedge operations, explicit Gauss-period decomposition, or verification that each successive quadratic extension is realized by a constructible radical. This omission is load-bearing for the main result, as the degree-2^16 extension requires thousands of nested steps whose correctness cannot be checked from the given text.
minor comments (2)
  1. The abstract and introduction would benefit from a short outline of the 'new approach' before the historical summary.
  2. Standard references to the minimal polynomial of cos(2π/65537) or to known tables of Gauss periods for smaller Fermat primes are missing and would help situate the new method.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for recognizing the potential historical and technical interest of an explicit construction for the 65537-gon. We address the major comment below and will revise the manuscript accordingly to strengthen the presentation of our new approach.

read point-by-point responses

  1. Referee: The central claim that the construction of the regular 65537-gon is shown 'exactly and without gaps' (abstract and the dedicated analysis section) is not supported by any enumerated sequence of compass-and-straightedge operations, explicit Gauss-period decomposition, or verification that each successive quadratic extension is realized by a constructible radical. This omission is load-bearing for the main result, as the degree-2^16 extension requires thousands of nested steps whose correctness cannot be checked from the given text.

    Authors: We agree that a fully enumerated list of every individual compass-and-straightedge operation would be impractical to include in a journal article, given the scale of the 2^16-degree extension. Our manuscript instead presents a new systematic approach based on a recursive decomposition of the Gauss periods for the cyclotomic field Q(zeta_65537). This decomposition is described in the dedicated analysis section, together with the explicit sequence of quadratic equations that realize each successive extension in the tower. Each step is shown to be constructible by solving a quadratic whose coefficients lie in the previous field, thereby filling the gaps left by Hermes' unpublished work. We acknowledge, however, that the current text does not include a complete numerical verification of the first few layers or a tabulated period decomposition for all 16 levels. In the revised version we will add an explicit Gauss-period decomposition for the initial two layers, together with a verification that the corresponding radicals are constructible, and we will include a clear algorithmic description of how the remaining layers follow identically. This will make the gap-free character of the construction more readily verifiable while preserving the paper's focus on the new method rather than an exhaustive enumeration. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper builds its central claim on the independently established theorems of Gauss (1796) and Wantzel (1837) characterizing constructible regular polygons via quadratic extensions in cyclotomic fields. These are external, long-verified results in algebraic number theory, not re-derived or self-referenced within the present work. The new approach for n=65537 is described as an explicit sequence of compass-and-straightedge operations, with the claim of showing the construction 'exactly and without gaps' constituting the paper's purported independent content rather than a reduction to fitted inputs or self-citations. No equations, parameters, or self-definitional loops appear; historical references (including to Hermes) are to prior external announcements, not load-bearing self-citations by the author. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the standard theory of constructible numbers and the known list of Fermat primes; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Compass and straightedge operations generate exactly the constructible numbers (field extensions of degree power of 2).
    Invoked implicitly when stating that the 65537-gon is constructible.

pith-pipeline@v0.9.0 · 5801 in / 1087 out tokens · 39875 ms · 2026-05-22T13:34:38.903812+00:00 · methodology

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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