**God Has OCD**

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**Pythagorean Triples and Their Prime Factors**

**—**

**Sum Of Powers and Their Prime Factors**

The Pythagorean Theorem is familiar to all of us who ever took Geometry in High School. But there’s more to it than we may have realized:

**A**^{2}** + B**^{2 }**= C**^{2}

This post is about yet another project where we investigate whether (or not) the terms of the equation “share” prime factors. In this case we consider * Pythagorean Triples* where A, B, and C are

**all integers**(of course). It wasn’t the first time I had come across Pythagorean Triples but this time I decided it might make for a simple project to determine whether the terms of the

*“share” Prime Factors.*

**Pythagorean Theorem**We’ll expand / extend the findings to apply to “* Sum of Powers*” equations in general. More on that later.

This question of “sharing” and “non-sharing” of Prime Factors among the terms of formulas of other “famous” theorems was the subject of other posts in this blog. Here are links to those other posts that investigate the sharing (or non-sharing) of Prime Factors for other notable formulas/theorems:

EVEN = HALF the difference of 2 squares

ODD = The difference of 2 squares

Prologue To Gerry’s Goldbach Theorems

**Pythagorean Triples**

It is helpful to first read a little about Pythagorean Triples. Here is a link to an excellent and very interesting (yet short) treatment that’s well worth reading. It’s on **Wolfram.com**

http://mathworld.wolfram.com/PythagoreanTriple.html

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__Primitive__ Pythagorean Triples

In this post/project we only deal with __ Primitive__ Pythagorean Triples (where A and B are

__) because, as stated on the Wolfram page:__

**relatively prime**It is usual to consider only __ primitive__ Pythagorean Triples (also called “reduced”triples) in which A and B are relatively prime

**since other solutions can be generated trivially from the primitive ones.**

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**Step 1 – Generate Primitive Pythagorean Triples**

The first step of this small project was to generate a bunch of Primitive Triples and to write them on a whiteboard; and then to manually determine whether they share any Prime Factors with the variable C (on the right side of the equation).

Keep in mind we are only considering Primitive Triples. Therefore A and B will NOT “share” any Primes because, by definition, they don’t share ANY factors (prime or otherwise); so we’ll be looking to see whether it’s possible for A and B to “share” a prime with C (on the right side of the equals-sign).

As it turns out, generating the Triples was very easy because I already had written a C# program that would do that. It was/is **the “****SumOfPowers****” program** that finds integer solutions for “*Sum of Powers*” equations like:

T_{1}^{n} + T_{2}^{n} + .. T_{x}^{n} = Z^{n}

For this project I simply ran **the SumOfPowers program **specifying just 2 terms on the left side of the equation and an exponent value of 2. Like so…

A^{2 + }B^{2 }= C^{2}

In this way the * SumOfPowers* program could instantly generate MANY combinations of A, B, and C (all of them integers) which I could then easily factor to see if they share any Prime Factors.

Here are a few very simple (and small) examples of the values for A, B, and C found by the * SumOfPowers* program.

3, 4, 5

5, 12, 13

11, 60, 61

119, 120, 169

48, 55, 73

36531, 36540, 51669

Of course, **the ****SumOfPowers**** program** could generate gigantic quantities of other examples where A, B, and C varied tremendously in size. Note that the * SumOfPowers* program is the subject of its own article (post) that you can read here:

**http://itbegsthequestion.com/sumofpowers/**

**Step 2 – Manually Factor and Evaluate the Generated Examples**

There’s not much to say here. We just manually reduce each term (A, B, C) down to their Prime Factors and see if any of those Prime Factors show up in more than 1 term. As it turned out, for all of the 20 or so sample *Primitive* Pythagorean Triples examined, there was never a case where a Prime Factor of either A or B was also a Prime Factor of C. I.e. for any particular Triple there was never a Prime Factor that showed up in more than 1 of the terms A, B, or C!

The fact that Prime Factors of A and B never appeared as a Prime Factor of C led to the next step which was to see if we could prove that this would **always** be the case.

**Step 3 – Prove It**

Here’s the proof (short and simple).

Let A, B, and C represent a __ Primitive__ Pythagorean Triple in the following equation (note that we simply put C on the left and put A and B on the right):

**C**^{2 }**=** **A**^{2 }**+** **B**^{2 }

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Show the above with C’s Prime Factors P_{c1}^{..}P_{cn} exposed:

**(P**_{c1}** x P**_{c2}** x .. P**_{cn}**) x C = ** **A**^{2 + }**B**^{2 }

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Divide both sides by one of C’s factors (e.g. P_{c1}) and we get…

**(P****c2**** x .. P****cn****) x C = A**^{2}** + B**^{2 }

** —– ——-**

** P**_{c1} **P**_{c1 }

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**.**

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If either A or B is to “share” a Prime Factor with C, then P_{c1}

1– Must divide ONLY A or ONLY B, or it

2– Must divide BOTH A and B.

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**.**

1– If P_{c1} divides just 1 term (ONLY A or ONLY B ) then we’d have the following (e.g. with P_{c1} dividing ONLY A):

Integer = Integer + B^{2} / P_{c1}

Integer = Integer + Non-Integer

Integer = Non-Integer <<<========== contradiction

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**.**

**2– ** If P_{c1} divides BOTH A and B then A and B are NOT relatively prime and therefore A, B, and C did NOT constitute a *primitive* triple after all which contradicts our original assumption/premise.

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**.**

**Therefore, C can NOT “share” a Prime Factor **

**with either A or B.**

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Pretty simple and straightforward, right? So what does it all mean?

Well, it should be obvious…

It means that *God exists and has OCD.** *

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It seems like we may be able to **apply the above proof to Sums Of Powers in general**. Instead of just having 2 terms on the left and one term on the right and an exponent of 2 like the Pythagorean Theorem:

A^{2 }+ B^{2 }= C^{2}

We can get more general with any number of terms and any exponent. Like so:

T_{1}^{n} + T_{2}^{n} + .. T_{x}^{n} = Z^{n}

That will form the basis for the next project. I’ll let you know if it gets anywhere.

**Open The Gates!**

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