Construction of Regular Prime p-Gon Exists iff p is Fermat Prime
Theorem
Let $p$ be a prime number.
Then there exists a compass and straightedge construction for a regular $p$-gon if and only if $p$ is a Fermat prime.
Proof
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Also see
- Proposition $1$ of Book $\text{I} $: Construction of Equilateral Triangle
- Proposition $11$ of Book $\text{IV} $: Inscribing Regular Pentagon in Circle
Historical Note
The result Construction of Regular Prime $p$-Gon Exists iff $p$ is Fermat Prime was stated, but not proved, by Carl Friedrich Gauss, who demonstrated the result for $n = 17$ in $1796$, when he was $18$.
The case $p = 257$ was demonstrated by Magnus Georg Paucker in $1822$, and again by Friedrich Julius Richelot in $1832$.
The case $p = 65 \, 537$ was attempted by Johann Gustav Hermes, who offered a construction in $1894$ after a decade of work. However, it has been suggested that there are mistakes in his work.
The cases where $p = 3$ and $p = 5$ were known to the ancient Greeks and are given in Euclid's The Elements.
Sources
- 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $17$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $257$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $17$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $257$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Fermat number (P. de Fermat, 1640)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Fermat number (P. de Fermat, 1640)