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Old 2016-07-19, 14:33   #1
bhelmes
 
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Default pyth. triples and primes

A peaceful day for all,

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 :

http://devalco.de/System/system_one_complex_pyth2.php

Perhaps there is somone who likes this explication.
Suggestion and improvements are welcome.

Nice greetings from the primes
Bernhard
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Old 2016-07-19, 15:31   #2
R. Gerbicz
 
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"Robert Gerbicz"
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Quote:
Originally Posted by bhelmes View Post
A peaceful day for all,

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.
Define your relationship. For which (a,b,c) Pythagorean triple the image will be the (0,1) point on the (unit) circle?
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Old 2016-07-20, 10:34   #3
bhelmes
 
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A peaceful day

Quote:
Originally Posted by R. Gerbicz View Post
Define your relationship. For which (a,b,c) Pythagorean triple the image will be the (0,1) point on the (unit) circle?
I think for (a,b,c) the trivial solution (0,1,1) is a Pythagorean triple
(not very interesting) and the image will be the (0,1) point on the unit circle.

I will add this commentar on the page because it is not clearly expressed.

Nice greetings from the primes
Bernhard
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