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Cyclotomy

In the last section of the Disquisitiones Gauss proves that a regular n -gon can be constructed with straightedge and compass if φ (n) is a power of 2. If 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 n is a first power and n − 1 is a power of 2. The primes that are one more than a power of 2 are called Fermat primes, 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 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 n are