# 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

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## 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)