Editing Fermat number (section)

==Relationship to constructible polygons==
[[File:Constructible polygon set.svg|thumb|300px|Number of sides of known constructible polygons having up to 1000 sides (bold) or odd side count (red)]]
{{Main|Constructible polygon}}

[[Carl Friedrich Gauss]] developed the theory of [[Gaussian period]]s in his ''[[Disquisitiones Arithmeticae]]'' and formulated a [[sufficient condition]] for the constructibility of regular polygons. Gauss stated that this condition was also [[necessary condition|necessary]],<ref>{{cite book |last1=Gauss |first1=Carl Friedrich |title=Disquisitiones arithmeticae |date=1966 |publisher=Yale University Press |location=New Haven and London |pages=458–460 |url=https://archive.org/details/disquisitionesar0000carl/ |access-date=25 January 2023}}</ref> but never published a proof. [[Pierre Wantzel]] gave a full proof of necessity in 1837. The result is known as the '''Gauss–Wantzel theorem''':

: An ''n''-sided regular polygon can be constructed with [[compass and straightedge]] if and only if ''n'' is either a power of 2 or the product of a power of 2 and distinct<!-- Define "distinct" --> Fermat primes: in other words, if and only if ''n'' is of the form {{nowrap|1=''n'' = 2<sup>''k''</sup>}} or {{nowrap|1=''n'' = 2<sup>''k''</sup>''p''<sub>1</sub>''p''<sub>2</sub>...''p''<sub>''s''</sub>}}, where ''k'', ''s'' are nonnegative integers and the ''p''<sub>''i''</sub> are distinct Fermat primes.

A positive integer ''n'' is of the above form if and only if its [[Euler's totient function|totient]] ''φ''(''n'') is a power of 2.