The reason that Beatles records sounded just like the Beatles is that the inverted stamper (that stamped out all those vinyl records) contained all the patterning (in inverse form) of the original (positive) record cutting.
The reason that the Goldbach conjecture can be verified, is that the set of all primes numbers contains (in inverse form) all of the patterning of the set of prime multiples. The equidistant patterning of prime multiples imposes (stamps) upon the primes the equidistant patterning necessary for endlessly successful Goldbach pairing.
For instance, in order for the even number 20 to have a successful Goldbach pair, two prime numbers equidistant from 10 must be found (say 13 and 7) both of these numbers are 3 numbers in distance from the mid point 10. This equidistant patterning is provided by the prime multiples (of 2 and 3).
"Multiples" are by definition equidistant or "evenly spaced." The result is that all prime number-cluster positions (the spaces between prime multiples) are likewise equidistant/evenly spaced. Being evenly spaced they are perfect for Goldbach pairing.
With the exception of "1" all numbers are either primes or multiples of primes. The pattern of prime multiples is subsequently perfectly transferred to the primes.
Within the system, there is no source of Goldbach defeating non-equidistance.
4 comments:
Your stronger conjecture of equidistance is merely a rephrasing of Goldbach's original conjecture.
When you state that any number larger than three will have two distinct equidistant primes you state that for any number n larger than three there exist at least two primes p and q so that n - p = q - n.
n - p = q - n
n - p + (n + p) = q - n + (n + p)
n + n - p + p = q + p - n + n
n + n = p + q
2n = p + q
2n is by definition an even number and so this is the original Goldbach Conjecture.
I would just like to explain my Goldbach article a little more. In particular the “targeting plank” of the proof. In order for the Goldbach conjecture to be true, the location of primes needs to be targeted for successful pairing.
The number system can be divided into 3 sets. The set of all prime multiples (4,6,8,9 etc), the set of all primes (2357 etc,) and the small set consisting of only the number one (1) - the only number that is neither prime nor a multiple of a prime. Being mutually exclusive and exhaustive, the pattern established by the set of all prime multiples is necessarily reflected (as a photographic negative) in the positioning of the set all primes. Neither of these sets is random. The validity of the Goldbach conjecture is dependent upon comprehensive prime positioning (targeting) for Goldbach pairing – plus prime availability due to system finiteness for all finite evens (see original article).
As the photographic-negative image (of the set of all prime multiples), the set of all primes is utterly non-random and just as mathematically predictable as the set of all prime multiples. Just as prime multiples are factorable so the primes are “intervalled.” To use an analogy, just as a vinyl record is grooved then so is the initial stamper that produced it, ridged. The ridges of the stamper necessarily contain all of the music (the form) albeit in an inverted state. Just as the set of all prime multiples is composed of evenly spaced multiples so is the set of all primes composed of evenly spaced intervals. In music, the “interval” (of say, 4th or 5th) is a relative term and not anchored to any particular key. Prime intervals are not anchored to zero (as prime multiples are) but are instead out of phase or “co-anchored” to zero. Successful Goldbach pairing requires even number intervals between primes.
While new even number multiples (and subsequent subordinate intervals) are being produced at a linear rate (by adding two to each even number {2468..etc}), ongoing interval drop-outs (due to uneven prime multiples) occurs at only a square root rate (see article). The result is that the Goldbach conjecture is true (for all finite even numbers).
Just to let you know, my article has been published at this site..
http://unsolvedproblems.org/
..and can be discussed at the adjacent chat forum.
Cool!
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