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Re: Interpretations of Goldbach's Conjecture
Posted:
Apr 9, 2000 6:15 AM
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Two observations here:
1. In your bidirectional sieve, the result _has_ to be symmetrical because
what you've done to the string is symmetrical. Also, notice that your
bidirectional sieve appears to be the rephrasing of your prime symmetry
around any integer.
2. Your folded primes, clearly, are also a restatement of your prime
symmetry around any integer. The reason you get a non-zero result from a*b
is because n+m and n-m will both appear in the same column.
I haven't looked carefully at your table intersections, but I'm already
getting a glimmer that that, too, may be but another consequence of your
prime symmetry.
-Doug Magnoli
Derek Ross wrote:
> Hello,
>
> Here are some different interpretations of Goldbach's Conjecture.
> They've probably been done already(?).
>
> I'm not sure why I'm posting this, it's more like I'm thinking out loud.
> I don't solve anything, but maybe it'll stimulate some interesting
> converstion on the topic...
>
> I've included in this posting four different interpretations of the
> conjecture (yes, the names are of my own creation).
>
> 1) PRIME SYMMETRY AROUND ANY INTEGER
> 2) BIDIRECTIONAL SIEVE
> 3) FOLDED PRIMES
> 4) GOLDBACH TABLE INTERSECTIONS
>
> ------------------------------------------------
>
> 1) PRIME SYMMETRY AROUND ANY INTEGER
>
> For any integer N > 2, there exists a value M such that
>
> N + M is a prime
> N ? M is a prime.
>
> 2) BIDIRECTIONAL SIEVE
>
> Take a sequence of integers, 1 to N where N is an odd number. In this
> example I'll use N=13
>
> 1 2 3 4 5 6 7 8 9 10 11 12 13
>
> First, go through the array from 1 to N, and eliminate all non-primes.
>
> - 2 3 - 5 - 7 - - - 11 ? 13
>
> Next perform a sieve in reverse direction, from N down to 1, and remove
> all non-primes, assuming that N represents 1 and 1 represents N.
>
> - - 3 - - - 7 - - - 11 - -
>
> By Goldbachs Conjecture, such an operation will
> A) always leave at least one number after the bidirectional sieve is
> completed,
> B) leave a symmetrical distribution.
>
> 3) FOLDED PRIMES
>
> Take an odd sized sequence of integers and fold it down the middle. For
> this example, I'll use N=13.
>
> 1 2 3 4 5 6 7
> 13 12 11 10 9 8 7
>
> Now, replace all primes with a 1 and all non-primes with a 0:
>
> 0 1 1 0 1 0 1
> 1 0 1 0 0 0 1
>
> By the Goldbach Conjecture, if row A is ANDed with row B, the result
> must be non-zero.
>
> A : 0 1 1 0 1 0 1
> B : 1 0 1 0 0 0 1
> A*B : 0 0 1 0 0 0 1
[snip]
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