It Begs The Question
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Gerry’s Goldbach Weak Theorem December 24, 2015

Trippin’ Over Goldbach… Again.

BaniaSeinfeldNoCaption

That’s Goldbach Jerry!  Goldbach!

~~~~~~~~~~~~~~~~~~~~~~~~

Last night I had another bout of insomnia, but not the kind where you can’t fall asleep; I always fall asleep within a handful of minutes of my head hitting the pillow. Rather, it was the kind where you wake up in the middle of the night (sometimes multiple times) and have a difficult time getting back to sleep. That’s the way it was last night. Up at 1:45 AM… Listen to CoastToCoast with George Noory to see what they were talking about.

NooryPhoto

George Noory (Looks nothing like his voice)

Last Night’s show was not very interesting. It was about large Mt. Rushmore sized faces carved into the sides of valleys… ON MARS!! So I turned off the radio but still could not get back to sleep. Instead,

Lock The Gates!

Instead I started thinking about… Now Focus! And don’t let me see those eyes roll!…

Instead I started thinking about:

The Odd (or “weak“) Goldback Conjecture:

Every odd number greater than 5 can be expressed as the sum of three primes (A prime may be used more than once in the same sum.). That is:

G1 + G2 + G3 == Odd

where Odd  is any odd number   >   5   and

G1 , G2 , and G3 are Prime Numbers

This conjecture is called “weak” because if Goldbach’s strong conjecture (concerning sums of two primes) is proven, it would be true. (Since if every even number greater than 4 is the sum of two odd primes, merely adding 3 to each even number greater than 4 will produce the odd numbers greater than 7.)

Also see the previous article on this blog titled Gerry’s Goldbach Theorem which covers the strong Goldbach Conjecture and associated Theorem.

 

Hey… It’s a cheap hobby.  You don’t even need shoes for it!

 

Anywho0,  I started following the line of thought found in Gerry’s Goldbach Theorem regarding the Prime Factors of E (the EVEN number) for Goldbach’s strong conjecture:

G1 + G2 == E (for E > 2)

From Goldbach’s Strong Conjecture. Where E is even integer

and G1 and G2 are Primes (aka “Goldbach Primes“).

Specifically, I thought it was “puzzling” that. for the “strong” conjecture shown just above, that the Prime Factors of E would NEVER appear as one of the Goldbach Primes (G1 and G2 ) except under one   very specific circumstance.

And with that in mind, but instead considering the Odd (weak) Goldbach Conjecture, I started working out some solutions (examples) in my head. Surprisingly enough, “solutions” to the Odd/Weak Conjecture appeared to be the “opposite” (or “mirror”) of the Proven solution to the Even/Strong Conjecture!!!

That is, it appeared that for :

G1 + G2 + G3 == Odd       (Goldbach’s Weak Conjecture)

EVERY Prime Factor of Odd  will ALWAYS appear

at least once as a Goldbach Prime (G1 , G2 , or G3 )

Except for the very specific case where Odd  itself is a Prime.

 

This time, however, instead of immediately contemplating a computer program to provide additional “evidence” via BILLIONS or TRILLIONS of examples I decided that a Proof might be much easier; especially since a Proof for Gerry’s Goldbach Theorem came about relatively easy (for the “strong” conjecture).

 

And indeed it was easier. Here it is (just below). It ain’t Rocket Surgery and it probably won’t change the election… but still you may find it interesting.

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

G1 + G2 + G3 == Odd

From Goldbach’s Weak Conjecture where Odd is an odd integer

and G1, G2 and G3 are Primes (aka “Goldbach Primes“).

————————————————–

.

.

Odd  ==   P1 x P2 x P3.. x Pn

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

————————————————–

.

.

Choose any one of the Prime Factors of   Odd  (e.g. Px). Then

Odd  –  Px == E       

Since Px is oddE is then some EVEN number

————————————————–

.

.

E == G1 + G2

From Goldbach’s Strong Conjecture.

The Gs are Primes that we can actually compute at this point!

————————————————–

.

.

Odd  ==    E + Px    ==     G1 + G2    +    Px

Odd  ==   G1 + G2    +   Px

————————————————–

And we can do the above for each and every

Prime Factor Px  of Odd  

.

In other words

Odd  numbers   >   5 are the sum of 3 primes

and EVERY Prime Factor of the Odd  number

is one of the Goldbach Primes

for at least one Goldbach Partition

.

EXCEPT FOR THE CASE where Odd is itself Prime

because Px would equal Odd thus making G1 and G2 both zero (but zero is not Prime)

End of Proof  – End of Proof  – End of Proof  – End of Proof  – End of Proof  – End of Proof  – End of Proof  – End of Proof  – End of Proof  – End of Proof  – End of Proof  – End of Proof  – End of Proof  – End of Proof  – End of Proof  – End of Proof  –

💡 Ta Da! 💡 

Of course, all of the above is predicated on Goldbach’s Strong Conjecture being TRUE…

The Truth Is Out There

We just have to wake up in the middle of the night to find it.

 

The End

3AM_WeakGBNotes_015-12-24

Original Notes made about 3:00 AM on 12/24/2015

on what was handy at the time (a sheet of paper towels).

Click on it to view full-size.

Open The Gates!

 

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