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GOD Has OCD October 7, 2016

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God Has OCD

Pythagorean Triples and Their Prime Factors

Sum Of Powers and Their Prime Factors

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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:

A2 + B2 = C2

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 Pythagorean Theorem “share” Prime Factors.

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

Gerry’s Goldbach Theorem

Gerry’s Goldbach Weak Theorem

Chen’s Theorem and YAPFO

 

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 relatively prime) because, as stated on the Wolfram page:

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:

T1n + T2n + .. Txn = Zn

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…

A2 + B2 = C2

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):

C2 = A2 + B2

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Show the above with C’s Prime Factors Pc1..Pcn exposed:

(Pc1 x Pc2 x .. Pcn) x C = A2 + B2

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Divide both sides by one of C’s factors (e.g. Pc1) and we get…

(Pc2 x .. Pcn) x C =    A2    +    B2

                                     —–            ——-

                                Pc1              Pc1

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If either A or B is to “share” a Prime Factor with C, then Pc1

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

2–  Must divide BOTH A and B.

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1–  If Pc1 divides just 1 term (ONLY A or ONLY B ) then we’d have the following (e.g. with Pc1 dividing ONLY A):

Integer = Integer +     B2 / Pc1

Integer = Integer + Non-Integer

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

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2–  If Pc1 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.

ocdmeme

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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:

A2 + B2 = C2

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

T1n + T2n + .. Txn = Zn

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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