Number Theory

    

Fermat Primes to Become Criterion for the Constructibility of Regular 2^k-Sided Polygons

Authors: Pingyuan Zhou

Abstract: Gauss-Wantzel theorem shows that regular n-sided polygons, whose number of sides contains a(distrinct) Fermat prime(s) as odd prime factor(s) of n or number of sides is power of 2, are all constructible with compass and straightedge. But of these caces, the constructibility of all regular 2^k-sided polygons is not related to Fermat primes. We discover the number of so-called root Mersenne primes Mp for p4 ( though there are no known Fermat primes Fk for k>4 ) then the constructibility of all regular 2^k-sided polygons can be indirectly explained by Fermat primes as criterion. Thus there exist direct or indirect connections between Fermat primes and all constructible regular polygons according to the theorem.

Comments: 9 Pages. Author gives an argument for indirect connections between Fermat primes and regular 2^k-sided polygons to make Gauss-Wantzel theorem have general sense in implying connections between Fermat primes and all constructible polygons.

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Submission history

[v1] 2014-07-29 02:38:15

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