Construction of Regular Prime p-Gon Exists iff p is Fermat Prime/Historical Note
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Historical Note on Construction of Regular Prime $p$-Gon Exists iff $p$ is Fermat Prime
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): $17$
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.)
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.25$: Gauss ($\text {1777}$ – $\text {1855}$)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $17$
- 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)
- 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $7$: Patterns in Numbers: Gauss