24/01/19 11:54:09.75 cbeVFClI.net
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(参考)
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Seventeagon
Mathematical background
Gauss's discovery is based on a solution to the circular division equation
x^17-1=0 whose solutions – the seventeenth unit roots – form a regular seventeenth with radius 1 in the Gaussian number plane of complex numbers.
In 1796, as an 18-year-old, Gauss recognized this possibility: "Through strenuous reflection ... In the morning... (before I got out of bed)"[4] due to general number-theoretic properties of prime numbers, in this case specifically the prime number 17: The modulo of a prime number
p formed by 0 different residual classes
1,・・・ ,p-1 can be used as potencies
g^0=1,g^1=g,g^2,dotsc ,g^p-2 a suitably chosen number
g, called primitive root. In particular, in the case of
p=17 can be concretely
g=3 as a recursive calculation of the powers shows:
3^0=1, 3^1=3・1=3, 3^2=3・3=9, 3^3=3・9 mod 17=10, 3^4=3・10 mod 17=13,
5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6
If you now sort the 17th unit roots of 1 different according to the order, i.e. in the order
ζ , ζ ^3, ζ ^9, ζ ^10, ζ ^13, ζ ^5, ζ ^15, ζ ^11, ζ ^16, ζ ^14, ζ ^8, ζ ^7, ζ ^4, ζ ^12, ζ ^2, ζ ^6,
thus, by partial summation of every second, every fourth, or every eighth unit root from this list, one obtains the so-called Gaussian periods: two 8-membered periods with 8 summands each, four 4-membered periods with 4 summands each, and eight 2-membered periods with 2 summands each. On the basis of fundamental properties or through explicit computation, it can be shown:[5]
・The two 8-part periods are solutions of a quadratic equation with whole coefficients.
・The four 4-part periods are solutions of two quadratic equations whose coefficients are calculable from the 8-part periods.
・The eight 2-part periods are solutions of four quadratic equations, the coefficients of which are calculable from the 4-part periods.
For the two-part period to the "first" unitary root,
ζ +ζ ^16=ζ +ζ ^-1=2cos(2π /17).
The described approach can be analogously applied to any prime number of the form
2^2^k+1 carry out. Five such prime numbers, called "Fermat's primes", are known: 3, 5, 17, 257, 65537. Therefore, the regular 257 vertex and the regular 65537 vertex are also among the constructable polygons.
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