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For 2000 years it had been known that regular polygons with

3, 6, 12 …sides;

4,8, 16 … sides;

5,10,20 … sides and

15, 30, 60 …sides

could be constructed with only a straight edge and compasses. That left undetermined if polygons with 7, 9, 11, 13, 14, 17 or 19 sides could be constructed. In 1796, Gauss, then 19, announced that a 17-sided polygon could be constructed AND that none of the others could be. He also showed that polygons with the number of sides equal to the product of certain prime numbers could be constructed, so polygons of 3 x 17 and 5 x 17 sides could be constructed. The series of primes was 2,3,5,17, 257, 65537 all of which can be written as

FermatPrimes

So far, all higher Fermat numbers are composites
F5 = 2**32 + 1 = 4,294,967,297 === 641 × 6,700,417 (found by Euler)
F6 = 2**64 + 1 = 18,446,744,073,709,551,617 === 274,177 × 67,280,421,310,721 (found by Thomas Clausen; more on him later)

For 5 ≤ n ≤ 32 all are composite BUT complete factorizations of Fn are known only for 0 ≤ n ≤ 11, and there are no known prime factors for n = 20 and n = 24. As far as I know, there is no proof that there are no more Fermat primes.

 

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