Given the spectral analysis from the previous post, we may notice that the presence of [1,1] is relevant to the GC. Potential absence of [1,1] for an even number would disprove the conjecture. Still, my computational research did not allow disproving the conjecture.
Still, I would like to pinpoint a couple of interesting observations (shown as graphs):
1. [1,1] (sum of two numbers of dimension 1)
2. [2,2](sum of two numbers of dimension 2)
3. [1,2](next numbers that can be shown as sums of numbers of dimensions 1 and 2 respectively)
4. odd numbers numbers that cannot be shown as sum of two primes
In [1,1] and [2,2] cases similar situation. Guessable. Unless I’m missing something.
[1,2] – next numbers that can be shown as sum of numbers of 1 and 2 dimensions (respectively). We conjecture its specific growth. Nothing special. Guessable, ie. the same gradient should work for all [a,b] for distinct integers a,b, only shifted.
Imagine Goldbach would say that every number (not only the even ones) can be shown as sum of two primes. Odds that would not “do it” would look like this. Nontrivial of those are
11 17 23 27 29 35 37 41 47 51 53 57 59
65 67 71 77 79
I’m calling it the GD (Goldbach Dislikes) and think learning more about this formula might help me understand the GC better, ie. why it’s all about the even numbers and not all numbers. It’d be for all numbers hadn’t it been for the GD. In the chart below we see its linear approximation. The differences between those numbers are as follows:
6 6 4 2 6 2 4 6 4 2 4 2 6 2 4 6 2, but I don’t have enough data and understanding of the subject to hazard a likely conjecture to pursuit (as for today).
Additionally, I would like to show you something I’d been thinking of for a while.