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October 3rd, 2017, 10:53 AM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 371 Thanks: 26 Math Focus: Number theory  # of Pythagorean ntuples
For n > 1, what count of Pythagorean ntuples tends to be greatest?

October 3rd, 2017, 12:20 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,539 Thanks: 591 
What is being counting? Number of triplets is infinite.

October 3rd, 2017, 04:19 PM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 371 Thanks: 26 Math Focus: Number theory 
I guess they all belong to sets that are uncountable, i.e. relatively the same size?

October 3rd, 2017, 06:59 PM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 395 Thanks: 211 Math Focus: Dynamical systems, analytic function theory, numerics 
An interesting fact is that during the course of the deriving the generating relations for all pythagorean triples one proves a bijection between triples and ration numbers. Thus, they are countably infinite. The basic idea is a fun enough exercise. Start with $x^2 + y^2 = z^2$. Divide through by $z^2$ so that a triple corresponds to a point on the unit circle. Now find any 1 solution to this equation and realize that any line with a rational slope passing through your solution will intersect the circle once more in another solution. From here one can get explicit generating relations for generating all pythagorean triples. 
October 3rd, 2017, 08:47 PM  #5 
Member Joined: Jul 2015 From: tbilisi Posts: 30 Thanks: 8 
$\displaystyle {x_1}^2+{x_2}^2+...+{x_n}^2=y^2$ $\displaystyle x_1=2{p_1}s$ $\displaystyle x_2=2{p_2}s$ ..... ...... ...... $\displaystyle x_{n1}=2{p_{n1}}s$ $\displaystyle x_n={p_1}^2+{p_2}^2+....{p_{n1}}^2s^2$ $\displaystyle y={p_1}^2+{p_2}^2+....+{p_{n1}}^2+s^2$ Last edited by Individ; October 3rd, 2017 at 08:50 PM. 

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