2017-08-10 09:45:20 UTC
Now, I do not recall when I started with the Staircase conjecture, a conjecture original to me-- was it 5 years ago, that I started this? Do not recall but can go back and find out in the post date time headings.
Anyway, what the Staircase Conjecture is, it is the reverse of Goldbach in that Goldbach wants to decompose even numbers into prime addition. Staircase wants to build even numbers starting with a toolbox of primes and adding onto existing primes. Once you see the shemata below, you instantly recognize what is going on::
Caution: I am using 1 as Unit Prime
1+1 = 2
1+3 = 4
1+5 = 6
3+5 = 8
5+5 = 10
7+5 = 12
1+13 = 14
3+ 13 = 16
5 + 13 = 18
7 + 13 = 20
3 + 19 = 22
Now let me stop there and point out a iteration, that I strive to start each iteration with 1 so I can use the 1,3,5,7 pattern, so I want that pattern to appear as often as possible, and it was temporarily broken with 22. but starts right back up again at 24
1 + 23 = 24
3+ 23 = 26
5 + 23 = 28
7 + 23 = 30
Now, let me stop there and state the Conjecture I have in mind, in full.
STAIRCASE CONJECTURE:: Given a bag of primes as tools, generate all Even Numbers in as best of a pattern as possible, using 1 in that bag, and, I conjecture that in the Grid Systems of Mathematics that each prior Grid has sufficient primes as the toolkit to generate all the Even numbers up to the next higher Grid.
So, what do I mean? I mean in the example started above for which I will post all the even up to 100, that the primes in the 10 Grid which are 1, 2, 3, 5, 7, and omitting 2, that those primes suffice to build all the evens from 0 to 100. Now, the conjecture implies that all the primes in 0 to 100 (omit 2) suffice to build all the evens from 0 to 1000, and so on so forth in higher Grids.
What this means is that the Prime gaps are never so large that previous Grid primes can span that gap.
In 10 Grid the largest gap was 8,9,10. In 100 Grid, the largest prime gap was from 89 to 97 for a gap of 8 and it tested the conjecture with 7+89 = 96. So if the number 89 were not prime but instead no primes from 83 to 97, then the Staircase Conjecture would be instantly wrong.
Now I have to review the largest prime gap span in 1000 Grid, but I am confident that the primes in 100 Grid easily span the gap.
Now, there are many interesting questions about the maximum order we can assign the building. We would want as many as possible of runs of 1, 3, 5, 7 for that gives us immediately four even numbers in a row. And we ask the question, is the number of those runs involve pi or 2.71... So looking at my data sheet, I count up 7.5 runs of 1,3,5,7 and I count 5 runs of 3,5,7. So that would be 30 of the 50 even numbers are of the largest run.
So now, I need data from the 1000 Grid, to see if pi or 2.71... is involved.
And, perhaps we can spot a pattern in primes.
But a cool interesting feature of the Staircase Conjecture, is that if true, it instantly proves Goldbach, for it is the reverse of Goldbach.