 # Development ofAlgorithmic Constructions

 12:49:34 19.Sep 2021

### 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 vectorin the circle of unit 1 ---> group of complex elements |a+bI|=1 mod p with a,b element Np ---> Mersenne numbers

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

7 31 127
zahl = 2^5-1

 Number p=

### p = 31

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. 1 0 1 -> (0,1), (1,0), (0,30), (30,0) 4 , 1. 1. 3 4 5 -> (13,7), (18,7), (13,24), (18,24)(7,13), (7,18), (24,13), (24,18) 8 27, 27 1 2. 2. 5 12 13 -> (2,11), (29,11), (2,20), (29,20)(11,2), (11,29), (20,2), (20,29) 8 7, 13 1 3. 3. 7 24 25 -> (4,4), (27,4), (4,27), (27,27) 4 0, 1 1 0. 0. 3 10 10.5 -> (0,0), (0,0), (0,0), (0,0) 3 ,

 I 0 1