Trippin’ Over Goldbach… Again.
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.
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 odd, E 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.
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!