Since all prime numbers that are not 2 or 3 belong to either the 6n-1 or 6n + 1 sequences then we can divide
the Goldbach conjecture into 6 possibilities.
- (6n-1) + (6m-1) = 6n + 6m -2
- 3 + 6n+1 = 6n + 4
- (6n-1) + (6m+1) = 6n + 6m
- (6n+1) + (6m-1) = 6n + 6m
- (6n+1) + (6m+1) = 6n + 6m +2
- 3 + 6n-1 = 6n +2
This shows that even numbers divisible by 6 with 0 remainder having more possibilities for
being the sum of two prime numbers.
46 48 50
3+43 5+43 3+47
5+41 7+41 7+43
11+35 11+37 13+37
17+29 13+35 19+31
23+23 17+31 25+25
19+29
23+25
52 54 56
3+49 5+49 3+53
5+47 7+47 7+49
11+41 11+43 13+43
17+35 13+41 19+37
23+29 17+37 25+31
19+35
23+31
25+29
Here is a scatter chart going up to 30,000. The even group is represented by the green dots, the plus group by the red dots and the minus group by the blue dots.
On a whim I added a Line of Best Fit and was surprised to find that the lines for the minus (blue) and plus (red) were separate. I had expected them to be almost identically placed.
To make this easier to see, here is a scatter graph for the Comet from 24,000 to 30,000. The gap is very noticeable. I have no idea why it occurs.
The gap is still there from 900,000 to 900,200.
The gap is still there from 900,000 to 900,200.
In the course of playing around with the Goldbach Conjecture, I decided to see if there were any other patterns that might be interesting.
The above graph is a scatter chart going up to a bit less than 5000 which is designed to show in a proportional sense how each even number belonging to the 6n - 2 group may have one or more Goldbach solutions. The x axis is for each of these even numbers. The y axis shows the relative position of each Goldbach solution. I love the little twirls at the far left.
To show more detail, here is a version of the 6n-2 group (minus) going up to 1000.
The even number 10 has a solution of 5 + 5 so there is a dot for it at the 0.5 level.
The even number 22 has two solutions, 5 + 17 and 11 +11. There would be a dot at the 0.5 level for the 11 + 11 solution and also matching dots at the 5 / 22 (0.227) and 17/22 (0.773) level. By the way, I ignored the solutions involving 3 such as 3 + 19.
I found these gap lines fascinating and after a while I realized what was going on. The key is to first look at the bottom (almost horizontal) line of dots. It represents the prime number 5. Then look at the top most almost horizontal line which represents a 6n-2 number minus 5. A dot on this line means that the number is prime and when matched with the prime number 5, it satisfies the Goldbach criteria. If there is no dot, then the number is not prime and just a 6n-1 composite.
Once the presence or absence of a dot on these upper and lower boundary lines is established, then that presence or absence is reflected in the lines that initially run at roughly right angles to the boundary lines. However when a presence line matches with an absence line, the result is no dot.
I have found it helpful to think of the gaps as being created by the composite bulldozer. The bulldozer blade is at its widest as it starts off at a boundary and gradually narrows as it continues on forever.
I have also found it helpful to see the overall pattern as intersecting waves with peaks and troughs.
I found these gap lines fascinating and after a while I realized what was going on. The key is to first look at the bottom (almost horizontal) line of dots. It represents the prime number 5. Then look at the top most almost horizontal line which represents a 6n-2 number minus 5. A dot on this line means that the number is prime and when matched with the prime number 5, it satisfies the Goldbach criteria. If there is no dot, then the number is not prime and just a 6n-1 composite.
Once the presence or absence of a dot on these upper and lower boundary lines is established, then that presence or absence is reflected in the lines that initially run at roughly right angles to the boundary lines. However when a presence line matches with an absence line, the result is no dot.
I have found it helpful to think of the gaps as being created by the composite bulldozer. The bulldozer blade is at its widest as it starts off at a boundary and gradually narrows as it continues on forever.
I have also found it helpful to see the overall pattern as intersecting waves with peaks and troughs.
This graph also provides some insight as to how the lower boundaries of the Goldbach comet come about.
If you click on the above graph you will see a larger version of the graph with lower values at approximately 226, 286, 346, 550 and 790. If you look back at the proportional scatter graph for 1000, you will see that it is around these numbers on the x axis that the bulldozer starts off.
Since the bulldozer is headed almost at right angles to the x axis it carves out a large proportion of the available solutions the closer it is to its starting point. As the bulldozer gradually turns the corner and proceeds more horizontally, its proportional effect grows smaller.
It would appear that number of Goldbach solutions for a given even number is not totally probability driven but partially depends on how close it is to a sequence of composites and the length of that sequence of composites.
Since the bulldozer is headed almost at right angles to the x axis it carves out a large proportion of the available solutions the closer it is to its starting point. As the bulldozer gradually turns the corner and proceeds more horizontally, its proportional effect grows smaller.
It would appear that number of Goldbach solutions for a given even number is not totally probability driven but partially depends on how close it is to a sequence of composites and the length of that sequence of composites.
Here is the version for even numbers up to 1,000 belonging to the 6n + 2 group. In this case I just show the lower half so that you can see the pattern with more clarity.
Here is the version for even numbers up to 1,000 belonging to the 6n group. As expected, the density of the dots for the 6n version exceeds the densities for the 6n-2 and 6n+2 versions. However in all three cases, the same gap lines appear.
This version shows them all on the one graph. Note that the gap lines still exist.
Bertrand's postulate states that there is always one prime p such that n < p < 2n- 2. One other consequence of breaking the Conjecture into the groups 6n-2 (minus), 6n (even) and 6n+2 (plus) is that there must always be at least one prime number in the upper half of groups 6n-2 and 6n+2 that makes Goldbach true. Hence if Goldbach is true, then
n < p and q < 2n
where p belongs to the 6n-1 group and q belongs to the 6n+1 group of prime numbers.
Of course if Goldbach is true, there have to be multiple prime numbers in the upper half but that is another thing for me to puzzle about. How many primes do there have to be and how do the sequences of composites effect that number of primes.
The above graph gives a way of visualizing whether the Goldbach conjecture still works at very high values with large gaps between primes. It does this by constructing a fake prime sequence. In this case we are only looking at the (6n - 2) numbers up to 2000. The fake prime sequence consists of the 6n-1 prime numbers less than 500 and three 6n-1 prime numbers chosen at random, 641, 743 and 983.
What is interesting is that the conjecture does not work for this fake sequence, particularly above 1500. Consequently when working with the actual prime sequence, it looks difficult for the conjecture to be true when the gap between primes becomes huge.
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