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God Has OCD Part Deux November 8, 2016

God Has OCD – Part Deux

Did you know that a shortage of remainders is why the hypotenuse can’t have prime factors in common with the legs of a right triangle?  Interesting eh?  Want to know more?  Then read on.

Lock The Gates!

In part 1 of God Has OCD we dealt with the Pythagorean Theorem    A2 + B2 = C2

and we proved that, for primitive Pythagorean Triples, A and B do not ever “share” prime factors with C. That is, A and B do not ever have a prime factor that is also a prime factor of C.

In this article we’ll look into the “sharing” of prime fctors for the more general Sum Of Powers formulas of which the Pythagorean Theorem is merely a special case.

We can express the general Sum Of Powers formulas like so:

T1n + T2n + T3n .. + Txn = Zn

In the above we’ll call T1n + T2n + T3n .. + Txn the “terms” of the equation.

We’ll demonstrate how, for “primitive” N-tuples, where the greatest common denominator of all the terms is 1,  that the terms may or may not, “share” prime factors with Z; except for, again, the special case of the Pythagorean Theorem where prime factors are never “shared” and where there are only 2 terms (A and B as we’re accustomed to seeing them called).

So where do we begin? As I often do, I experimentally test theories first, and then see if they can be proven. In this case, I could again use my SumOfPowers program (in C#).

With the SumOfPowers program we can specify:

        • How many terms to use
        • A range of values for the terms
        • The exponent (n) to use

 

 

The program would then find integer solutions if any exist. Follow the link below for an example screen print of the SumOfPowers program after an execution.

SumOfPowers Example

For this article I ran the SumOfPowers program and the results demonstrated that sometimes the terms T1..Tx will have a prime factor in common with Z and sometimes none of the terms will have a prime factor in common with Z.

That said, we now unequivocally know and can demonstrate the answer with a real example but the question is why? And then, why is this NOT true for the case of 2 terms (e.g. the Pythagorean Theorem)?

Anyway here are a few simple examples found by the SumOfPowers program:

T13 + T23 + T33 .. + T63 = Z3

13 + 23 + 33 + 63 + 163 + 173 = 213

13 + 23 + 63 + 63 + 63 + 73 = 103

33 + 43 + 53 = 63

23 + 53 + 123 + 123 + 153 + 173 = 243

T15 + T25 + T35 .. + T85 = Z5

85 + 145 + 165 + 205 + 205 + 225 + 245 + 305 = 345

 

 

So why is it possible for the Sum Of Powers for more than 2 terms to share prime factors with Z while it is NOT possible when there are just 2 terms?

T1n + T2n .. + Txn = Zn            << YES prime factors sometimes shared!

T1n + T2n = Zn                     <<  NO.  Less than  3  terms then Prime factors NEVER shared!

 

 

The reason is as follows… If we divide both sides of the equation by one of Z’s prime factors we get something like this (for example):

33 + 43 + 53 = 63                               Prime factors of 6 are 3 and 2. Let’s use 3…

33      +      43      +      53      =        63

—-             ——           ——-            ——–

3                 3                3                 3

 

 

3 divides 3^3 and it also divides 6^3

however (4^3)/3 and (5^3)/3 are not integers but the equation is still valid because

The sum of all of the remainders is an integer multiple of 3. For example:

(4^3)/3 = 21 + 1/3 and

(5^3)/3 = 41 + 2/3 then

1/3 + 2/3 = 3/3 = integer

So it all works out because we still get integers on both sides of the equation.

But note that this could never work out when there’s only 2 terms because then there would only be a single remainder after dividing the terms T1 and T2 by one of the prime factors of Z. And with only a single remainder, the left side of the equation can’t possibly be an integer!

For example, for A2 + B2 = C2

let’s assume that one of the prime factors of C was also a prime factor of A. Let’s call that common prime factor Pc. Then we would have something like the following if we divide both sides of the equation by Pc:

(Pc x Pa2 … x Pan)  x A       +         B2        =         (Pc x Pc2 … x Pcn) x C

——————————                    —————               ————————————-

        Pc                                           Pc                                    Pc

 

Integer + ( B2 / Pc) = Integer

But    B2 / Pc        yields some remainder after division; And

 

Unlike the case of more than 2 terms on the left side of the equation,

there are no additional remainders possible to add to this remainder

in order to obtain an integer that would make the equation valid!!!

 

Integer   +    ( B2 / Pc)        =     Integer

Integer   +   NON-Integer    =     Integer            <<=== a contradiction

 

 

So our original assumption of some prime factor of C also being a prime factor of A can NOT ever be true for the Pythagorean Theorem or for any Sum Of Powers formula with just 2 terms on the left of the equals sign.

 

 

And in some quirky and curious sense

we can say that this is because,

for just 2 terms (like the Pythagorean Theorem),

there will always be a “shortage of remainders”

that would prevent the validity of the equation!!!

All done.  We can go now.

Open The Gates!

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