Inhaltsverzeichnis

Development of
Algorithmic Constructions

16:30:56
Deutsch
20.Apr 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

31 127 511
zahl = 2^7-1

Number p=

p = 127

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),
(0,126), (126,0)
4,
1.1. 345 -> (26,50), (101,50),
(26,77), (101,77)
(50,26), (50,101),
(77,26), (77,101)
881, 601
2.2. 51213 -> (59,40), (68,40),
(59,87), (68,87)
(40,59), (40,68),
(87,59), (87,68)
8103, 211
3.3. 72425 -> (46,60), (81,60),
(46,67), (81,67)
(60,46), (60,81),
(67,46), (67,81)
840, 591
4.4. 94041 -> (25,30), (102,30),
(25,97), (102,97)
(30,25), (30,102),
(97,25), (97,102)
8106, 1031
5.5. 116061 -> (21,24), (106,24),
(21,103), (106,103)
(24,21), (24,106),
(103,21), (103,106)
8119, 1191
6.6. 138485 -> (39,2), (88,2),
(39,125), (88,125)
(2,39), (2,88),
(125,39), (125,88)
8120, 291
7.7. 15112113 -> (8,8), (119,8),
(8,119), (119,119)
40, 11
0.0. 106061 -> (0,0), (0,0),
(0,0), (0,0)
7,

I
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