Inhaltsverzeichnis

Development of
Algorithmic Constructions

13:16:04
Deutsch
19.Sep 2021

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odd number < 2000 :
0 + i
^20 + i
^20 + i1 + i
^20 + i1 + i4 + 5i3 + 4i3 + 10i
^20 + i1 + i3 + 4i5 + 12i8 + 15i10 + 11i
^20 + i1 + i3 + 4i1 + 2i2 + 3i1 + 4i2 + 5i


Prime / (2 * prim.pyth.trippel)
= 31 / (2*6) =
2.5833333333333
3.1415926535898 (= Pi)

The complex number are calculated by exponentation from the botton to the top modulo the input number.

The factorisation depends of the number modulo 4:
Let p be an odd prime, then
for p = 1 mod 4 the factorisation depends on p-1
for p = 3 mod 4 the factorisation depends on p+1

The calculated complex number a+bi depends on the tangens(alpha)= a/b
which describes a rational point on the unit circle.

For symmetric reasons the complex number a+bi is choosen for a < b,
otherwise the tangens(alpha)=a/b is replaced by the cotangens(alpha):=b/a,
that means a+bi:=b+ai for a > b.

If a and b have a common factor with div=gcd (a,b), then a:=a/div and b:=b/div.
(This means that the angle is the same but not the norm (a, b)=a²+b²)

if a=0 then b:=1.

The red elements a+bi are non quadratic residues and have jacobi (a²+b², f)=-1