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Some trials on the Goldbach Conjecture

Animation of ranges starting at 3 size 400 with 200 range iterations (next p(403…803) and so on) of necessary and first occurence of even number by adding 2 primes, going down pn+p(n…m) and then right p(n..m)+pm going on by pn+1 at next smaller prime iteration.

This one is for the first 400 primes > 2000000001

As you may see the uniquness grows and it becomes more an more a complete triangle. (s. 2 below)

2008-11-19 GBV seems still to be open

(Today I started this; But this is not chronological)

Some days ago I messed around with adding prime numbers instead of counting lambs to go asleep. Before it worked, I rememberd, that this has to do with the Goldbach Conjecture, and I decided to do some computer trials the next day.

I’m a naive mathematican, so formal strength is missing here. But although I have not the math means to prove any interssting thing (ok, this conjecture seems to be hard for others too), I think the Goldbach Conjecture is intuitive evident, as opposed to may be Riemans Conjectur, that’s only intuitive evident for some people (I am blind on this level).

So, this is only my fun in thinking about things. And if someone could give me some hint’s why many of my thoughts are wrong, which is probably the case, I would be very pleased, if you mail me some notes. ( But if you think, that this thoughts are in any way inventive, let me also know.

From now on I talk probably in most cases (not all, but that’s not the point) about evens >=6 and P={3,5…}. pn=prime number with index n a.s.o.

1 Searching for patterns

I wrote some programms to find the smallest, biggest and middlest prime number such that pn+pm=x, with x%2=0 and of course a gb-function that finds all 2-partitions with primes for a given even number. But at first look: no patterns.

I found that if want to test many numbers it is much simpler and faster to generate and hash the even numbers by adding pn+pm for all pn and pm with m<=n.

Because differences of evens and of primes > 3 are allways multiplies of 2 my first trial was to generate a triangle that marks the primes by walking down step 2 starting at a list of primes, which gives a sort of triangle (biger version by clicking on the picture):

Ok there are patterns, but I think they are not too interssting, although this is a representation of the complete number theorie (the first vertical row is the list of prime numbers).

1.1 Distribution of Ways

2008-11-20: I found a new pattern (click on it to get a big version):

While thinking about the fact (I found a prove by Leslie Green at wikipedia) that n primes generate at least n+n-1 different even numbers and Leslie Green stated that there is no pattern visable (see animation and picture at top of this page), the idea came up, to look how the distribution of multiple ways is when you take n prime numbers. This means: If you take the first n prime numbers (y-axis) an generate even numbers, the plot shows how many primes are used only by the unique (even numbers that have only one pair of primes that generate them) generatings, over the two way evens up to the maximum way (x-axis) evens that could be generated by the first n prime numbers. Example: The plot shows, that there are some even numbers that could be generated by the first 2000 primes in 517 ways, where 1034 different primes are involved. It also shows, that there are some even numbers, where only one way exists to generate them, and hereby only 31 different prime numbers are involved.

So, would you expect a surprise if we go deeper in the prime universe? (Or is this completely stupid? Which may be, because I am only interessted in fun ;-) But prime numbers are in the top 100 of my fun. (Ther is a top 1000, ... too)


I managed to get bigger Prime sets with the Rabin-Miller test. (Cause of its probalistic nature, I can not be shure that the sets are 100% right, but I think they are) So I could do 2 more like the pictures at the top of the page now with the first 400 prime numbers in the range of 10e19 an 10e38.

So I think, there is an interessting problem to solve:

If we take connected ranges of primes of size n. Is there a function, that is defined for all n, such that f(n)=p. with [p(x),...,p(x+n)] generating n(n+1)/2 different even numbers by adding 2 of them?

For example:

f(1)=2 cause [2] generates 4 and

f(2)=3 cause [3,5] generate 3 even numbers [6,8,10]

which are the trivial cases.

f(3)=5 cause [5,7,11] generate [10,12,14,16,18,22]

but this seems not to go on in this trivial sense, as we see at the the triangle pictures here and on top of the page. But as we can also see, there is a tendency to the triangle, if go further out into the prime universe.

2008-12-07 (2 hours later):

Here the results for the first 500000000 Primes:

1 [2]

2 [3, 5]

3 [5, 7, 11]

4 [19, 23, 29, 31]

5 [113, 127, 131, 137, 139]

6 [509, 521, 523, 541, 547, 557]

7 [2551, 2557, 2579, 2591, 2593, 2609, 2617]

8 [19267, 19273, 19289, 19301, 19309, 19319, 19333, 19373]

9 [75041, 75079, 75083, 75109, 75133, 75149, 75161, 75167, 75169]

10 [753143, 753161, 753187, 753191, 753197, 753229, 753257, 753307, 753329, 753341]

11 [4310819, 4310827, 4310851, 4310897, 4310927, 4310947, 4310963, 4310989, 4311001, 4311007, 4311011]

12 [38142197, 38142233, 38142241, 38142289, 38142331, 38142389, 38142409, 38142413, 38142439, 38142473, 38142527, 38142541]

13 [307379717, 307379731, 307379767, 307379773, 307379789, 307379791, 307379837, 307379869, 307379899, 307379983, 307380011, 307380023, 307380049]

to go further will get hard, cause you need special hole between primes constellations where the easiest is: no repeats (each holesize only one time in a possible pattern), but also complexer ones which I do not understand yet. And hole size and non repeating ranges size are very very slowly increasing. But infinity is far far away, so I think the first fitting 400 range (to get a nice triangle picture) must be ridiculously near.

1.2 Ulamizing the Prozess

I’ve tried to visualize the prozess of iterating over pn, pm. somthng like throwing the prime number fishernet from pn (3,5…) and show at every step the evens generated until pn.

Here two animated gifs:

1. The Ulamversion (spiralized, cause I think this is nicer)

2. The Straight version

1.3 Walking Primes

Another trial to find patterns. Start with a prime and add itself, if result +/- 1 is a prime go on adding first prime to the new one.

3 (8): [3, 5, 7, 11, 13, 17, 19, 23]
5 (8): [3, 7, 11, 13, 17, 19, 23, 29]
7 (9): [3, 11, 17, 19, 23, 29, 31, 37, 43]
11 (3): [3, 13, 23]
13 (9): [3, 17, 29, 31, 41, 43, 53, 67, 79]
17 (7): [3, 19, 37, 53, 71, 89, 107]
19 (8): [3, 23, 41, 43, 59, 61, 79, 97]
23 (1): [3]
29 (5): [3, 31, 59, 61, 89]
31 (1): [3]
37 (3): [3, 41, 79]
41 (3): [3, 43, 83]
43 (5): [3, 47, 89, 131, 173]
47 (1): [3]
53 (1): [3]
59 (2): [3, 61]
61 (1): [3]
67 (4): [3, 71, 137, 139]
71 (2): [3, 73]
73 (1): [3]
79 (4): [3, 83, 163, 241]
83 (1): [3]
89 (1): [3]
97 (7): [3, 101, 197, 199, 293, 389, 487]
101 (2): [3, 103]

Ok, may be I should delete Walking Primes but someway it brought the idea for the following.

... patterns may be continued, but now for the nice things:

2 Is there a more generalized Goldbach Principle?

I tried some more things with adding primes. I found for all primes in the range up to 10000000 every even number >= 3(pn-1) can be generated by adding 2 primes >= pn for all pn in the first third of this range and I think this goes on quite a while, if not for ever. May be there is even a smaller bound, anywhere between 2(pn+1) and 3(pn-1). For pn=3 this is the Goldbach Conjecture, because 2pn can always be generated.

May be someone could at least prove that my conjecture is wrong.

I know, that it’s easy to miss a thing when we talk about infinity, but for every range of prime numbers there ar so many more even numbers (yes not unique, that is just the sparkeling point) generated then there are in the range up to 2pmax.

We have n log n primes up to n, this means (n / log n)((n / log n)+1)/2 additions of two primes and only n even numbers up to 2 n. The 2 pictures on top of this page show that there is a strong tendency that uniqness of the generated evens grows with the ranges of primes. Ranges means here [p2,...,pn], [pn+1,...,p2n], [p2n+1,...,p3n],...

... may be a plot of 2n and (n / log n)((n / log n)+1)/2

After thinking about this, I came up with the holes between the primenumbers, because I think, the growth and especially the distribution of the holes is one key to prove the conjecture.

3 Beyond Twins

As far as I know, it’s not yet proven that there are infinitly many prime twins. I think this is also intuitive evident. I think further, that there are not only infinitly many twins of sort 2 (2-twins), this is p(n+1)-p(n)=2, there are also infinitly many 2n-twins, this means 4-twins (p(n+1)-p(n)=4), of course 6-twins (p(n+1)-p(n)=6), which are the most and so on.

I did a trial with the primes up to 1023893771, and this is what i got:

It seems that there is a progression for counting the holes between prime numbers that goes with Ci(x/log(x))/log(x/log(x)) with max(Ci) ~ 2.233 for the 6-twins and Ci2 ~ 1.255 for the 2-twins. And min(Ci) should be greater 0 for every even number. So this looks like all this progressions go to infinity.

This is not (klick on it to go to wikimedia):

but there seems to be connection.

Here a plot for Ci [2,4,..80]:

And I think there are also patterns, but overlaying. There is a 2 exp x pattern, there is a (2×3)xn pattern (the magic number 6) and there is a (2×5)xn pattern. So look at 30, this is an overlay of the 6 and the 10 pattern, so it does not fit to 6 or 10. I can not find a function, but i think some thing like gm(x) * log(x)/x comes near. with a gm(x) depending on the magic numbers (2xxn, 2×3, 2×5).

Ok, this all may look quite different in the range of googolplex to googolplexplex or greatest thinkable number by a human being (gtnbahb) to 2gtnbahb, but this is also a sparkeling point.

4 Folding the prime number fisherman’s net

As I told you in 1.2, I tried to find patterns by visualizing the prozess of iterating over pn, pm. So I had the following picture in mind: You have a fisherman’s net and throw it in the number sea. To catch all evens you first have to stand on a stone which is 3 high and throw a net that has just one stitch of size 3, 5,... and so on. Cause you got not every even number, you climb the next stone, which is 5 high, gather the 3 stitch an throw again. If Goldbach is right, after some steps you will catch all evens up to a certain n.

Thinking about this picture, a new Idea came up. What happens if we have to gather more stitches? The size of all is different, but we want to gather them compact, i. e. one stitch upside down, the next downside up and so on.

If we do this on a table? on which side would be more material at the end of gathering? The animation after the headline shows the progress of this hole-between-primes function for the first 250000 primes or so.

If done it over the first 999999999 prime numbers and got 7703 crosings of the x-axis with minimum value -185751 and maximum value 79107.

So, what has this to do with the Goldbach Conjecture? I’dont know, but I have a feeling, it has and it was fun to write the programm, that did the diagramms with gnuplot controlled by Python.

Fisherman Strategie II (added ad 2009-02-04)

My fisherman has an new Idea. “I have a net with this strange stiches (size 2, 3, 5 and so on). So what happens, if I go up a stairway and I take 2 steps every time and every time throw it back. I stand on an even numbered step and the knots of my net hit odd numbered steps. (ok the first one, which is at 2 does not, but this does not matter). So if I mark the knots that hit a prime numbered step, I will get all pairs of prime numbers that add to the step I’m standing on.

Here the result (up to step 250 and the steps are centered, cause the pattern comes mor visable) with one extra information, the red dots are even numbers of the form 2p, with p is prime, cause we do not have to prove the goldbach conjecture for this numbers:

and up to step 1600:

As one can see at the first picture, there are nested quadratic structures standing on the corners along the center top down line and the corners are all red i. e. mark evens that have the form 2p. And these red corners mark diagonal laps. If there are laps there are borders. So the borders show that has to be a dot at every line (even number). But follows this from the Goldbach Conjectur or a much stronger presupposition or follows the GB out of this lap border property (which it self may be follows from the 2p property and (p(n)-p(n-1))^2 is the hypotenuse of a right-angled triangle at whichs right angle corner must be a pair of prime numbers that generate the number p(n)-p(n-1), which is an even that has not the form 2p. So I think if we walk this rigthangle way through the laps starting at every red dot at the center line, we must come to the conclusion that there is a point at every horizontal line i.e. 2n. By the way: Every pair of odd primnumbers constructs a right angeled triangle with one new corner on the lap borders.

Hole Patterns (added 2008-11-29)

I checked the hole patterns of size 2 up to 1029999999. This is part of the result (only hole patterns up to 40-40, cause otherwise wen can not see anything):

So, what do I see. I think that there are many (how many? all?) patterns that will come forever. Candidate 1 is the 6-6 hole pattern. Ok 2 patterns come only one time 1-2 and 2-2.

... will be continued an changed. (For example: What’s so special about 6?)