## Fermat numberA number defined by the formula F_{n} = 2^{2^n}
+ 1 and named after Pierre Fermat who conjectured,
wrongly, that all such numbers would be prime. The first five Fermat numbers,
F_{0} = 3, F_{1} = 5, F_{2}
= 17, F_{3} = 257, and F_{4} = 65,537,
are prime. However, in 1732, Leonhard
Euler discovered that 641 divides F_{5}.
It takes only two trial divisions to find this factor because Euler showed
that every factor of a Fermat number F with _{n}n
greater than 2 has the form k × 2^{n+2} + 1.
In the case of F_{5} 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.
## Related categories• TYPES OF NUMBER• NUMBER THEORY | |||||

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