Inhaltsverzeichnis

Development of
Algorithmic Constructions

03:25:47
Deutsch
22.Jul 2019

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

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. 101 -> (0,1), (1,0),
(0,30), (30,0)
4,
1.1. 345 -> (13,7), (18,7),
(13,24), (18,24)
(7,13), (7,18),
(24,13), (24,18)
827, 271
2.2. 51213 -> (2,11), (29,11),
(2,20), (29,20)
(11,2), (11,29),
(20,2), (20,29)
87, 131
3.3. 72425 -> (4,4), (27,4),
(4,27), (27,27)
40, 11
0.0. 31010.5 -> (0,0), (0,0),
(0,0), (0,0)
3,

I
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