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Fermat number

A number defined by the formula Fn = 22^n + 1 and named after Pierre Fermat who conjectured, wrongly, that all such numbers would be prime. The first five Fermat numbers, F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65,537, are prime. However, in 1732, Leonhard Euler discovered that 641 divides F5. It takes only two trial divisions to find this factor because Euler showed that every factor of a Fermat number Fn with n greater than 2 has the form k × 2n+2 + 1. In the case of F5 this is 128k + 1, so we would try 257 and 641 (129, 385, and 513 are not prime). It is likely that there are only finitely many Fermat primes.

Gauss proved that a regular polygon of n sides can be inscribed in a circle with Euclidean methods (e.g., by straightedge and compass) if and only if n is a power of two times a product of distinct Fermat primes.

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