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November 6th, 2015, 01:34 PM   #1
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New Goldbach Conjecture foundation

(Cut & Paste to Notepad to get columns to line up)

Goldbach Conjecture Proof (almost)
1. For all primes > 5 there exist positive 0< m<=k integers such that p1<p2 can be expressed as p1=6(m)+1 or p1=6(m)-1 and
p2=6(k)+1 or p2=6(k)-1
2. So the sum of any two primes can be expressed as one of the below 4 equations.

p1 p2 p1+p2 2(n) n r=Goldbach radius
----------------------------------------------------------------------------
a. 6(m)+1 + 6(k)-1 = 6(m+k) = 2(3m+3k) => n=3(m+k) => r=(3(k-m)-1)

b. 6(m)+1 + 6(k)+1 = 6(m+k)+2 = 2(3m+3k+1) => n=3(m+k)+1 => r=3(k-m)

c. 6(m)-1 + 6(k)-1 = 6(m+k)-2 = 2(3m+3k-1) => n=3(m+k)-1 => r=3(k-m)

d. 6(m)-1 + 6(k)+1 = 6(m+k) = 2(3m+3k) => n=3(m+k) => r=(3(k-m)-1)

e. 6(k)+1 - 2(r) = p1 (Can be used where p1=3)

f. 6(k)-1 - 2(r) = p1 (Can be used where p1=3)

3. Except for 2, any two primes add to an even integer. Note: p1+p2 add to an even integer in all 4 equations.
4. 2(n) can express any even integer >= 10 and "n" can express any integer >= 5 looking at all 4 equations
5. From Bertrand's Postulate we know at least one p2 exists in n < p2 < 2n (I suspect many for large n)
6. From the Prime Number Theorem we know at least n/log(n) p1 primes exist in 0 < p1 < n.
7. Chose any even number >= 10. Chose m and k such that a.,b.,c. or d. is satisfied above. (a. and d. may both work)
8. Now choose the prime numbers that match the row(s). Goldbach is satisfied.
9. To disprove this, simply find "n" such that a.,b.,c. or d. cannot produce prime numbers, i.e. p1+p2 != 2(n)
Note: Integers 4, 6, 8 can be proved by hand.
10. If this does not lead to a proof, it may at least lay the groundwork for a proof since it deals with all the
elements of Goldbach including multiple partitions.

For Example:
2(n) = 22, so n=11 (prime) Already solved, but let's find more.

a./d. 6(m+k) = 2(3m+3k) = 2(n) => n=11 => 11=3m+3k => 11/3 = m+k (Impossible - Integers can't add to fraction )

b. 6(m+k)+2 = 2(3m+3k+1) = 2(n) => n=11 => 11=3m+3k+1 => 10/3 = m+k (Impossible)

c. 6(m+k)-2 = 2(3m+3k-1) = 2(n) => n=11 => 11=3m+3k-1 => 12 = 3(m+k) => 4=m+k => m=1,k=3;m=2,k=2
22 22
p1=5, p2=17, r=6
p1=11, p2=11, r=0
p1=3, p2=19, r=8 works for p2 of b.& c. but otherwise fails. But 19 - 2(r) = 3

2(n) = 18, so n=9 (odd but not prime)
a./d. 6(m+k) = 2(3m+3k) = 2(n) => n=9 => 9=3m+3k => 3=m+k => m=1,k=2

b. 6(m+k)+2 = 2(3m+3k+1) = 2(n) => n=9 => 9=3m+3k+1 => 8/3 = m+k (Impossible)

c. 6(m+k)-2 = 2(3m+3k-1) = 2(n) => n=9 => 9=3m+3k-1 => 10/3 = m+k (Impossible)
18 18
p1=7,p2=11,r=2
p1=5,p2=13,r=4
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November 12th, 2015, 07:11 AM   #2
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"5. From Bertrand's Postulate we know at least
one p2 exists in n < p2 < 2n (I suspect many
for large n)"

I don't think (n, 2n) suffices. Seems you need
a smaller interval. {I worked that out many,
many years ago but I could have been wrong
then - which of course means I could be
wrong now.}
_________
"I know just enough math to be dangerous."
(Anne N. Onymus)
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November 12th, 2015, 07:34 AM   #3
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Quote:
Originally Posted by rlrandallx View Post
7. Chose any even number >= 10. Chose m and k such that a.,b.,c. or d. is satisfied above. (a. and d. may both work)
OK... but a, b, c, and d are all true for all choices of m and k.

Quote:
Originally Posted by rlrandallx View Post
8. Now choose the prime numbers that match the row(s). Goldbach is satisfied.
What makes you think that you can do this? It seems like you tried to prove the Goldbach conjecture by assuming the Goldbach conjecture.
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November 13th, 2015, 08:50 AM   #4
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Bertrand's postulate, also called the Bertrand-Chebyshev theorem or Chebyshev's theorem, states that if n>3, there is always at least one prime p between n and 2n-2. Equivalently, if n>1, then there is always at least one prime p such that n<p<2n. (Wolfram)
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November 13th, 2015, 09:04 AM   #5
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Yes, I think we're all familiar with that -- what is your point?
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November 13th, 2015, 09:11 AM   #6
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For a. n=3(m+k) => 3,6,9,12,15, ... set A
b. n=3(m+k)+1 => 4,7,10,13,16 ... set B
c. n=3(m+k) -1 => 5,8,11,14,17, ... set C
d. n=3(m+k) => same as a.

n can only be in set A or D, B, or C so m and k are considered at most twice.
If 2n=32 and n=16, p2 = 25 if m=1 and k=4 but then p2 !=prime so you must select
another m,k pair so that n=16, e.g. m=2,k=3. Thus p1=13, p2=19. GC satisfied.
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November 13th, 2015, 09:21 AM   #7
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Yes, I've confused myself with this problem
many times and finally concluded that if 'at
least one prime in (n, 2n)' sufficed then the
the Conjecture would have been long-ago
proved.
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November 13th, 2015, 10:00 AM   #8
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There seem to be communication difficulties. Would you walk me through an example? Let's say I chose 2n = 3325581707333960528.
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November 14th, 2015, 04:09 AM   #9
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There is a big fundamental error in this proof.

Yes, you can write an even number in one or more of those forms. Yes, all primes > 5 can be written as 6k - 1 or 6k + 1. You have to remember 6k - 1 and 6k + 1 aren't always prime. Just because there exists a prime between n and 2n, all that means is you can find one prime of the form 6k - 1 or 6k + 1. It doesn't mean that the corresponding number (or second term in the equation) will be prime. You basically assumed what you were trying to prove. I'm not sure how clear that sounded, but I tried my best.
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