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March 7th, 2017, 08:24 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 281 Thanks: 24 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, 08:53 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,650 Thanks: 837 
they can't possibly be different triples and share two elements

March 8th, 2017, 07:44 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 10:14 AM. 
March 8th, 2017, 10:15 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,232 Thanks: 1437  
March 8th, 2017, 10:52 AM  #5 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 281 Thanks: 24 Math Focus: Number theory  Let's say either, as I am not sure what you mean.
Last edited by Loren; March 8th, 2017 at 11:17 AM. 
March 8th, 2017, 11:14 AM  #6  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,045 Thanks: 720  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 11:17 AM.  
March 8th, 2017, 06:28 PM  #7 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 281 Thanks: 24 Math Focus: Number theory 
Thanks to Euclid. Greater than his spacetime.

March 9th, 2017, 05:42 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 

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