It Begs The Question
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Gerry’s Goldbach Theorem December 12, 2015

For all positive EVEN integers E,

the Prime Factors of E are never one of E’s Goldbach Primes

except for the case where E divided by 2 is a Prime Number

(i.e. where E is “constructed” by simply doubling some prime).

GoldJerryGold

I had been looking for awhile for my next project. The project would ideally end up being a programming project based on something mathematical. But one can’t rush these things.  Patience… Let it bubble up in its own good time. The same for the “answer.”; in its own good time. All good things…

Anyway, during the waiting and after a couple of months I stumbled into Goldbach’s Conjecture. I don’t even remember how I stumbled into it but it was/is interesting so I mused over it off and on for a couple of weeks.   In retrospect it’s not surprising that I would “find” it since it’s notorious in the world of the history of mathematics. Just google “goldbach’s conjecture.”  So here’s the story…

 

Lock The Gates!

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Goldbach

Goldbach’s Conjecture is one of the oldest and best known unsolved problems in number theory and in all of mathematics. It states  –  Simply:

 

Every even integer greater than 2

can be expressed as the sum of two primes.

The conjecture has been shown to hold up through    4 x 1018   but remains unproven for over 250 years despite considerable notoriety and effort.  Here are some trivial examples:

5 + 3 == 8

7 + 7 == 14 and 3 + 11 == 14

5 + 13 == 18 and 7 + 11 == 18

Anywho… while playing with various numbers and manually calculating “Goldbach Primes” for easy cases I noticed that the Prime Factors of even numbers seemed to never be one of the Goldbach Primes; at least for the few examples I tried. Here is an example:

For the EVEN number 18

The “Goldbach Primes” are 5, 13 and 7, 11.

The prime factors are 2 and 3.

However, there is one exception to this “rule” of Prime Factors not being Goldbach Primes. And that is when the EVEN number divided by 2 is a prime.  Or, put another way, when the EVEN number is constructed simply by doubling some prime.   For example, consider the number 34:

34/2 == 17 (17 is prime)

One of the Goldbach Primes for 34 is 17 (17+17 == 34  thus 17 is, in some sense, the ONLY Goldbach prime)

The Prime Factors of 34 are 17 and 2

Thus the Prime Factor 17 is also a Goldbach Prime.

Thus my “conjecture”, which I call Gerry’s Conjecture, is what I came up with:

 

For all positive EVEN integers E,

the Prime Factors of E are never one of E’s Goldbach Primes

except for the case where E divided by 2 is a Prime Number

(i.e. where E is “constructed” by simply doubling some prime).

 

 

I performed numerous google searches and scanned many documents trying to determine whether anyone else stumbled into this. I could not find anything… of course, it may just be that I could not find any documentation. Anyway…

It was then that I realized what my next programming project would be;  it would be a C# program to calculate Goldbach Primes and Prime Factors for vast quantities of even integers and show which Prime Factors for an even integer are also one of the Goldbach Primes for that same integer… IF ANY!!! That was the plan.

I leisurely started outlining the program and its UI. This lasted a couple of days when I was called away to L.A. to help my father. While there I could not continue to work on the program but I did decide to play around with seeing if I could come up with a Proof. Oddly enough it occurred extremely quickly and I was very surprised that the proof wasn’t obvious very early on!

So the good news is that a proof was found.  A corollary was that there was then no point in developing the conjecture-testing project program (in C#).

So Gerry’s Conjecture is no longer a conjecture; it is now Gerry’s Goldbach Theorem (which, however, is based on the assumption that Goldbach’s Conjecture is true).

 

There are 8 million stories in the naked city...” and that was the one about Gerry’s Goldbach Theorem.

The Proof follows. Feel free to comment on it.  Thanks and enjoy.

 

 

 

Proof Follows – Proof Follows – Proof Follows – Proof Follows – Proof Follows – Proof Follows – Proof Follows – 

 

G1 + G2 == E  (for E > 2)

From Goldbach’s Conjecture.  Where E is even integer and  G1 and G2 are Primes. 

P1 x P2 x P3.. x Pn == E

From The Fundamental Theorem of Arithmetic (The Ps are primes)

 

 

G1 + G2    ==    E    ==    P1 x P2 x P3.. x Pn

G1 + G2 == P1 x N   where N == P2 x P3.. x Pn

(G1 / P1 ) + (G2 / P1) == N

Now let us assume that one of the Prime Factors (we’ll use P1) is one of the Goldbach Primes (we’ll use G2). That is, we’ll assume the opposite of what we are trying to prove and see if a contradiction occurs. So… if we let P1 == G2 we get the following…

(G1 / P1 ) + 1 == N

(G1 / P1 ) == N – 1    And remember, N == P2 x P3.. x Pn       and      P1 == G2 so that (G2 / P1) == 1

Note that N-1 is an integer!!!

 

So, N-1 is an integer but (G1 / P1 ) is NOT an integer so we reach a contradiction

EXCEPT FOR THE CASE G1 == P1

 

Therefore the only case where a Prime Factor is also a Goldbach Prime is the case where

E/2 == G1 == G2 == P1          I.E….     Gerry’s Goldbach Theorem

 

 

For all positive EVEN integers  E > 2,

the Prime Factors of E  are never one of the Goldbach Primes for E

except when E  divided by 2 is a Prime Number.

 

The End

 

 

 

 

 

 

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