
Number Theory Number Theory Math Forum 
 LinkBack  Thread Tools  Display Modes 
March 7th, 2017, 07:24 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 339 Thanks: 26 Math Focus: Number theory  Exclusive Pythagorean singles
Do there exist Pythagorean triples that do not share a member with any other triple? Or those that share two?

March 7th, 2017, 07:53 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,862 Thanks: 968 
they can't possibly be different triples and share two elements

March 8th, 2017, 06:44 AM  #3 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
Perhaps you are asking about Primitive Pythagorean Triples? If you are, then the answer is YES. You can see using Euclid's Formula which gives all Primitive Pythagorean Triples $ (a , b , c) = (m^2  n^2 , 2mn , m^2 + n^2)$ $ (3 , 4 , 5) = (2^2  1^2 , 2(2)(1) , 2^2 + 1^2)$ Look at $ a = 3$ , there is only one way to generate $3$ in Euclid's Formula (try it!) That fixes $m , n$ so $3$ cannot occur in any other triple. *** On the flipside, look at $c = 5$; there are only two ways to generate $5$ (try it!) and indeed there are only two Primitive Pythagorean Triples that have $5$ as a member. Last edited by skipjack; March 8th, 2017 at 09:14 AM. 
March 8th, 2017, 09:15 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,847 Thanks: 1568  
March 8th, 2017, 09:52 AM  #5 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 339 Thanks: 26 Math Focus: Number theory  Let's say either, as I am not sure what you mean.
Last edited by Loren; March 8th, 2017 at 10:17 AM. 
March 8th, 2017, 10:14 AM  #6  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 12,131 Thanks: 801  Quote:
3,4,7,8,9,11,16,19,23,27,31,32,43,47,49,59,64,67,7 1,79,81,83 Not 99? Nope. 2099101 and 9949004901 Last edited by Denis; March 8th, 2017 at 10:17 AM.  
March 8th, 2017, 05:28 PM  #7 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 339 Thanks: 26 Math Focus: Number theory 
Thanks to Euclid. Greater than his spacetime.

March 9th, 2017, 04:42 AM  #8 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
It is interesting to investigate a number like $15$ and see how many times it can be a member of a Pythagorean Triple (not necessarily primitive). Looking at Euclids Formula we see that there is only one way to generate 15. $ (a , b , c) = (m^2  n^2 , 2mn , m^2 + n^2)$ $ \ \ \ \ \ $ < Euclids Formula $ (15 , 112 , 113) = (8^2  7^2 , 2(8 )(7) , 8^2 + 7^2)$ $ \ \ \ \ \ $ < Primitive Pythagorean Triple We may be tempted to say $15$ occurs only once as a member. This is not the case however due to the fact that if $ (a , b , c ) $ is a Pythagorean Triple then so is $ ( ka , kb , kc ) $ for any positive integer $k$ Indeed , member $15$ can occur $2$ more times , genrated by the Primitive Pythagorean Triple $ (3 , 4 , 5 ) $ When $ k = 3 $ we generate $ (9 , 12 , 15 ) $ When $ k = 5 $ we generate $ (15 , 20 , 25 ) $ Member $15$ can also be generated by the Primitive Pythagorean Triple $ (5 , 12 , 13) $ when $ k = 3 $ giving $ (15 , 36 , 39) $ I found $4$ distinct Pythagorean Triples that have $15$ as a member. Are there more? *** There is a rich underlying structure here that depends on the total number of ways an integer can be written as the sum and/or the difference of $2$ squares. Whether a member is prime or composite (like $15$) also affects the total number of appearances in distinct Pythagorean Triples 

Tags 
exclusive, pythagorean, singles 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Can there be an exclusive and?  MadSoulz  Applied Math  2  August 31st, 2013 08:45 PM 
exclusiveor reduction  lamhmh  Applied Math  1  May 26th, 2011 11:42 PM 
Probability about mutually exclusive Events  hoyy1kolko  Probability and Statistics  1  April 11th, 2011 12:23 PM 
Events A,B (independent, mutually exclusive), Help!  Mark_jb  Advanced Statistics  1  February 23rd, 2011 01:16 PM 
question about exclusive/inclusive 'OR' in logic  pilfer00  Applied Math  1  June 7th, 2009 06:55 AM 