Inhaltsverzeichnis

Development of
Algorithmic Constructions
20:30:21
Deutsch
19.Apr 2024
1. Abstract
2. Transformation
3. Mathematical proofs
4. Symmetry
5. Program in pseudocode
6. Webprogram for the demonstration
  1. table with u and v
  2. table with calculation of the vectors from the pythagoraic triple
    3.
  3. picture for x^2+y^2=1 mod p
    4.
  4. table with the values
    5.
  5. picture for x^2-y^2=1 mod p
    6.
  6. table with the values
    7.
  7. picture with both values
    8.
7. Links

1. Abstract

There is a relationship between the primitiv Pythagorean triples (a,b,c) with a²+b²=c²
resp. the corresponding rational points of the circle with unit 1 (a/c)²+(b/c)²=1
and the group of the complex elements x²+y²=1 mod p where x, y are elements of Np.

In addition there is a similar relationship between the primitiv Pythagorean triples
resp. the corresponding rational points (c/b)²-(a/b)²=1 resp. (c/a)²-(b/a)²=1
and the group of the complex elements with x²-y²=1 mod p where x, y are elements of Np.

These relationships are one explication for the system of primes.

2. Transformation

Pythagorean
triples 		---> complex vector
in the circle of unit 1 		---> group of complex elements
|a+bI|=1 mod p with a,b element Np
Pythagorean
triples 		---> complex vector
sine hyperbola / cos hyperbola 		---> group of complex elements
a²-b²=1 mod p with a,b element Np

3. Mathematical proofs

    p an odd number > 1
    Let a,b,c a Pythagorean triple with c < p


a)    a²     + b²       = c²             by multiplication with (c²)⁻1
<=> (a/c)² +(b/c)²      = 1              with I²=-1
<=> (a/c)² -(bI/c)²     = 1



b)    a²     + b²       = c²             - b²
      a²                = c² -  b²       by multiplication with (a²)⁻1
<=> (c/a)² - (b/a)²     = 1              with I²=-1
<=> (a/c)² +(bI/c)²     = 1



c)    same caluclation, just -a² instead of -b²

    a²     + b²         = c²             - a²
                  b²    = c² -  a²       by multiplication with (b²)⁻1
<=> (c/b)² - (a/b)²     = 1              with I²=-1
<=> (c/b)² +(aI/b)²     = 1

4. Symmetry

The symmetry of the resulting group of the complex elements with norm=1
based on the commutativ structur of this group

a²+b²=b²+a²=c² resp.
(a/c)²-(bI/c)²=(b/c)²-(aI/c)²=1
and that x²+y²=(p-x)²+y²=x²+(p-y)²=(p-x)²+(p-y)² mod p;

Therefore one Pythagorean triple gives always 4 or 8 solutions with |x+YI|=1 mod p

The first trivial Pythagorean triple (0,1,1) gives the 4 solution (0,1), (1,0), (0,p-1) and (p-1,0).

If a Pythagorean triple has as result (x, y) with x=y then there are the following 4 solutions:
(x, x), (x, p-x), (p-x, x) and (p-x, p-x)

Otherwise there are 8 solutions for (x, y):
(x, y), (x, p-y), (p-x, y) and (p-x, p-y) and
(y, x), (y, p-x), (p-y, x) and (p-y, p-x)

5. Program in pseudocode

This is not a practical test for prime numbers, but a short description, how the webprogram works.
Let p an odd number > 1;
prime=True;
for i from all pyth. triples (a, b, c) with c < p
    if (gcd (c, prim)>1) then prime=False; break;
    calculation (a,b,c)-> (x, y);
    if (gcd (x, prim)>1) then prime=False; break;
    if (gcd (y, prim)>1) then prime=False; break;
    output of all elements of (x,y);
end_for;
if prime=True then p is prime;

6. Webprogram for the demonstration

p < 200 table of Pythagorean triples + graphic is shown + table with elements
p < 500 table of Pythagorean triples + graphic is shown
p < 10000 table of Pythagorean triples is shown

Number p=

p = 31

gcd (u,v)=1; u odd and v even, u even and v odd; u²+v² < 31

(2, 1);
(3, 2);
(4, 1); (4, 3);
(5, 2);


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

nr. u, v a b c -> (x,y) with |x+yI|=1 number of
elements		 factor ^n=1 (x,y) with x²-y²=1 number of
elements		 factor ^n=1
0.1,0 1=1²-0²0=2*1*01=1²+0² -> (0,1), (1,0),
(0,30), (30,0),
42 (1,0), (30,0),
21
1.2,1 3=2²-1²4=2*2*15=2²+1² -> (7,13), (18,7),
(13,24), (18,24),
(13,7), (7,18),
(24,13), (24,18),
88 (9,12), (22,12),
(9,19), (22,19),
(7,9), (24,9),
(7,22), (24,22),
88
2.3,2 5=3²-2²12=2*3*213=3²+2² -> (2,11), (29,11),
(2,20), (29,20),
(11,2), (11,29),
(20,2), (20,29),
816 (10,15), (21,15),
(10,16), (21,16),
(3,14), (28,14),
(3,17), (28,17),
832
3.4,1 15=4²-1²8=2*4*117=4²+1² -> (5,10), (21,5),
(10,26), (21,26),
(10,5), (5,21),
(26,10), (26,21),
816 (3,15), (28,15),
(3,16), (28,16),
(2,6), (29,6),
(2,25), (29,25),
82
4.4,3 7=4²-3²24=2*4*325=4²+3² -> (4,4), (27,4),
(4,27), (27,27),
44 (1,8), (30,8),
(1,23), (30,23),
41
5.5,2 21=5²-2²20=2*5*229=5²+2² -> (5,10), (26,10),
(5,21), (26,21),
(10,5), (10,26),
(21,5), (21,26),
016 (2,6), (29,6),
(2,25), (29,25),
(3,15), (28,15),
(3,16), (28,16),
016
32 30

number of primitiv pythagorain triples : 5
31 /(2*5) = 3.1

30 I
0 30

Number = 31, 31 = 3 mod 4

|0+30I|
=1
|11+29I|
=1
|20+29I|
=1
|4+27I|
=1
|27+27I|
=1
|10+26I|
=1
|21+26I|
=1
|13+24I|
=1
|18+24I|
=1
|5+21I|
=1
|26+21I|
=1
|2+20I|
=1
|29+20I|
=1
|7+18I|
=1
|24+18I|
=1
|7+13I|
=1
|24+13I|
=1
|2+11I|
=1
|29+11I|
=1
|5+10I|
=1
|26+10I|
=1
|13+7I|
=1
|18+7I|
=1
|10+5I|
=1
|21+5I|
=1
|4+4I|
=1
|27+4I|
=1
|11+2I|
=1
|20+2I|
=1
|0+1I|
=1
|1+0I|
=1
|30+0I|
=1
30 I
030

Number = 31, 31 = 3 mod 4

8²-30²
=1
23²-30²
=1
6²-29²
=1
25²-29²
=1
14²-28²
=1
17²-28²
=1
9²-24²
=1
22²-24²
=1
12²-22²
=1
19²-22²
=1
15²-21²
=1
16²-21²
=1
3²-16²
=1
28²-16²
=1
3²-15²
=1
28²-15²
=1
15²-10²
=1
16²-10²
=1
12²-9²
=1
19²-9²
=1
9²-7²
=1
22²-7²
=1
14²-3²
=1
17²-3²
=1
6²-2²
=1
25²-2²
=1
8²-1²
=1
23²-1²
=1
1²-0²
=1
30²-0²
=1
30 I
030

7. Links

Pythagoraen Triples by Chris K. Caldwell
Pythagoraen Triples, Wikipedia
Calculation of Pythagoraen Triples by Arndt Bruenner
Weisstein, Eric W. "Pythagorean Triple." From MathWorld--A Wolfram Web Resource.
List of Pythagoraen Triples with c < 10000, M. Somos,
A020882, Ordered hypotenuses (with multiplicity) of primitive Pythagorean triangles, Oeis
Pythagorean Right-Angled Triangles by R. Knott