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March 7th, 2017, 08:24 PM   #1
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Exclusive Pythagorean singles

Do there exist Pythagorean triples that do not share a member with any other triple? Or those that share two?
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March 7th, 2017, 08:53 PM   #2
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they can't possibly be different triples and share two elements
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March 8th, 2017, 07:44 AM   #3
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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.

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Last edited by skipjack; March 8th, 2017 at 10:14 AM.
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March 8th, 2017, 10:15 AM   #4
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Quote:
Originally Posted by Loren View Post
Do there exist Pythagorean triples that do not share a member with any other triple?
Did you mean a specific value or any value?
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March 8th, 2017, 10:52 AM   #5
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Quote:
Originally Posted by skipjack View Post
Did you mean a specific value or any value?
Let's say either, as I am not sure what you mean.

Last edited by Loren; March 8th, 2017 at 11:17 AM.
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March 8th, 2017, 11:14 AM   #6
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Quote:
Originally Posted by agentredlum View Post
$ (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.
Keeping short leg < 100, these are the ones that appear exactly once:
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. 20-99-101 and 99-4900-4901
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Last edited by Denis; March 8th, 2017 at 11:17 AM.
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March 8th, 2017, 06:28 PM   #7
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Thanks to Euclid. Greater than his spacetime.
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March 9th, 2017, 05:42 AM   #8
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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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