Inhaltsverzeichnis

Development of
Algorithmic Constructions

03:29:32
Deutsch
29.Mar 2024

1. Abstract

There is a relationship between the Pythagorean triples of the form a=2n+1, b=2n²+2n and c=2n²+2n+1
resp. the corresponding rational points of the circle with unit 1 in the complex field
and the Mersenne numbers especially the Mersenne prime numbers.

2. Transformation

subgroup of Pythagorian
triples
---> complex vector
in the circle of unit 1
---> group of complex elements
|a+bI|=1 mod p with a,b element Np
---> Mersenne numbers

3. Links

Gruppe der rationalen Punkte auf dem Einheitskreis
Group of rational points on the unit circle
Circle group
Cyclic group

1 7 31
zahl = 2^3-1

Number p=

p = 7

a, b, c is the Pythagorean triple with a²+b²=c²

nr. x a b c -> (x,y) with |x+yI|=1 factor number of
elements
^8 Distance
0.-1. 101 -> (0,1), (1,0),
(0,6), (6,0)
4,
1.1. 345 -> (2,2), (5,2),
(2,5), (5,5)
40, 11
0.0. 122.25 -> (0,0), (0,0),
(0,0), (0,0)
1,

I
01