Editing Euler's totient function (section)

===Cyclotomy===

{{main article|Constructible polygon}}

In the last section of the [[Disquisitiones Arithmeticae|''Disquisitiones'']]<ref>Gauss, DA. The 7th § is arts. 336–366</ref><ref>Gauss proved if {{mvar|n}} satisfies certain conditions then the {{mvar|n}}-gon can be constructed. In 1837 [[Pierre Wantzel]] proved the converse, if the {{mvar|n}}-gon is constructible, then {{mvar|n}} must satisfy Gauss's conditions</ref> Gauss proves<ref>Gauss, DA, art 366</ref> that a regular {{mvar|n}}-gon can be constructed with straightedge and compass if {{math|''φ''(''n'')}} is a power of 2. If {{mvar|n}} is a power of an odd prime number the formula for the totient says its totient can be a power of two only if {{mvar|n}} is a first power and {{math|''n'' − 1}} is a power of 2. The primes that are one more than a power of 2 are called [[Fermat prime]]s, and only five are known: 3, 5, 17, 257, and 65537. Fermat and Gauss knew of these. Nobody has been able to prove whether there are any more.

Thus, a regular {{mvar|n}}-gon has a straightedge-and-compass construction if ''n'' is a product of distinct Fermat primes and any power of 2. The first few such {{mvar|n}} are<ref>Gauss, DA, art. 366. This list is the last sentence in the ''Disquisitiones''</ref>
:2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40,... {{OEIS|A003401}}.