My Math Forum  

Go Back   My Math Forum > College Math Forum > Number Theory

Number Theory Number Theory Math Forum


Thanks Tree2Thanks
  • 2 Post By SDK
Reply
 
LinkBack Thread Tools Display Modes
October 3rd, 2017, 11:53 AM   #1
Senior Member
 
Joined: May 2015
From: Arlington, VA

Posts: 281
Thanks: 24

Math Focus: Number theory
# of Pythagorean n-tuples

For n > 1, what count of Pythagorean n-tuples tends to be greatest?
Loren is offline  
 
October 3rd, 2017, 01:20 PM   #2
Global Moderator
 
Joined: May 2007

Posts: 6,398
Thanks: 546

What is being counting? Number of triplets is infinite.
mathman is offline  
October 3rd, 2017, 05:19 PM   #3
Senior Member
 
Joined: May 2015
From: Arlington, VA

Posts: 281
Thanks: 24

Math Focus: Number theory
I guess they all belong to sets that are uncountable, i.e. relatively the same size?
Loren is offline  
October 3rd, 2017, 07:59 PM   #4
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 224
Thanks: 121

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.
Thanks from greg1313 and Loren
SDK is offline  
October 3rd, 2017, 09:47 PM   #5
Member
 
Joined: Jul 2015
From: tbilisi

Posts: 30
Thanks: 8

$\displaystyle {x_1}^2+{x_2}^2+...+{x_n}^2=y^2$

$\displaystyle x_1=2{p_1}s$
$\displaystyle x_2=2{p_2}s$
.....
......
......
$\displaystyle x_{n-1}=2{p_{n-1}}s$
$\displaystyle x_n={p_1}^2+{p_2}^2+....{p_{n-1}}^2-s^2$
$\displaystyle y={p_1}^2+{p_2}^2+....+{p_{n-1}}^2+s^2$

Last edited by Individ; October 3rd, 2017 at 09:50 PM.
Individ is offline  
Reply

  My Math Forum > College Math Forum > Number Theory

Tags
ntuples, pythagorean



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Prove That There are no 2-Tuples to Satisfy... James Brady Number Theory 0 December 2nd, 2016 11:25 AM
Pythagorean once more! Denis New Users 2 April 21st, 2013 10:02 PM
Formalization of tuples klendo Applied Math 1 March 12th, 2011 08:11 PM
Pythagorean Triples julian21 Number Theory 2 November 13th, 2010 01:01 PM
Pythagorean Triples II julian21 Number Theory 3 November 12th, 2010 12:58 PM





Copyright © 2017 My Math Forum. All rights reserved.