I’ve worked with the idea I had presented many posts ago, ie. with the spectrum of numbers and dimensions of numbers. Here, in this post, iteratively, I’m going to present you with the progress. Firstly, a new integer sequence has been defined. The Z-sequence (z-sequence ) for every integer includes a certain transformation of the amount of different dimension factors (e.g. [1,5],[2,2]) . When sketched, we see interval increase, very much resembling the one from the R-sequence as if it was the R-sequence which would define the amount of dimension factors.
A short analysis now. We don’t know all prime numbers. There is infinity of primes. If we choose (see: prime key set problem) n prime numbers, then we can use them to build all numbers that can be created with them. Geometrically, it means that we can n-dimensional (in the context of prime number goemetry introduced earlier) numbers using those primes. If our set of primes does not include a certain prime, then we will not be able to create numbers that divide our very prime. Thus, poetically, we will not be capable of hearing frequencies related to that prime. Any n-dimensional “polygon” built out of our chosen primes will not have any side-length that divides our prime, thus no resonant effect will take place. If we add yet another one prime to our prime number set, we will be able to create even more numbers, ie. we will be able to create all numbers that divide our newly added prime. Still, we will not be able to receive any signal from all the other primes that exist and don’t belong to our prime set.
From that we can clearly see that factorization allows us see “how much we know (can receive)”. Due to problems with factorization, we are not able to say “how much we know” all that often. And now, why do we use prime numbers for building numbers when we can easily use addition to create all the next numbers? The hypothesis is: we need to know more about number dimensions as those are the numbers that provide more information about the universe, with their geometry.
For e.g. [5,2] (so-called dimension factor), the bigger number is taken to define highest dimension number that can be added to another number to make a specific number. Thus,below I present how those high-dimension that make numbers from 1 to 200 look like. Again, hyperbolic or somewhat similar.
It’s even better shown by the graph of the second number’s dimension (from dimension factors for 1-200 numbers) (see below).
Still, believable, I guess, not the toughest sketch to conjecture. Looks like nothing changes differently to either R-sequence change or x/lg(x) approximation.
According to the re-stated Goldbach Conjecture, every even integer spectrum includes [1,1]. I analyzed the [2,2] case.D.S. McNeil pointed out to me that my software suffers from failure as e.g. 163 = 2*2 + 3*53 etc. I must work a bit harder on this one and won’t currently post these results.
Lets now think of the inclusion-exclusion method in terms of the former. It is claimed that every even integer has its spectrum with [1,1]. We notice that only certain even integers don’t include [2,2] and even more numbers don’t include [3,3]. Is it possible to define numbers that have [1,1], no [2,2], [3,3], no [4,4], [5,5], no [6,6] etc.? How do we know which numbers include [x,y] dimension factor in the spectrum?
Yet another question is whether we can notice anything important about numbers with spectrum that includes [p,p] for prime p. It is conjectured (the GC) that all even numbers have [1,1], but what numbers have [3.3]. Can we find a formula to find those numbers? my problems 2