A different way of looking a prime numbers

Most texts about prime numbers understandably look only at the prime numbers. This blog attempts to look at prime numbers but from a different point of view, the composites.

This blog will examine the following:

• How the prime sequences can be though of as two separate sequences

• The patterns of the unprime composites

• Some thoughts about 'twin primes'

• A Goldbach oddity and a different way of looking at Goldbach

At the end of this blog there are links to two other blog entries that follow on from this blog to examine the following:

How the prime series can be though of as two separate sequences

The first thing to notice about primes is that if the first two primes 2 and 3 are bypassed, all other primes belong to one of two sequences

            6n-1     example 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95….

            6n+1    example 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97….

Within each sequence, it is apparent that along with the primes there are also some composites, (35, 65, 77, 95….)     (25, 49, 55, 85, 91…..)

This is well known but usually the composites are then ignored for any further study. To differentiate these composites within either of the two sequences from the remaining composites divisible by 2 or 3 that are not part of the two sequences, let them be called ‘unprime composites’. Those that are part of the 6n-1 sequence I refer to as ‘minus unprime composites’ and those part of the 6n+1 sequence, I refer to as ‘plus unprime composites’.

If we look at the unprime composites in sequence 6n-1, it becomes apparent that they are all the product of a primes or composites from the 6n-1 sequence and the 6m+1 sequence.

            Eg        (6n+1) * (6m-1) = 36nm - 6n + 6m -1

If we look at the unprimes in sequence 6n+1, it becomes apparent that they are all the product of either two primes or composites from sequence 6n-1 or the product of two primes or composites from sequence 6m+1.

            Eg        (6n-1) * (6m-1) = 36nm - 6n - 6m +1

            or         (6n+1) * (6mb+1) = 36nm + 6n + 6m +1

For example,

            77 = 7 * 11 = (6*1 + 1) * (6*2 – 1)

            65 = 5 * 13 = (6*1 – 1) * (6*2 + 1)

            55 = 5 * 11 = (6*1 – 1) * (6*2 – 1)

            91 = 7 * 13 = (6*1 + 1) * (6*2 + 1)

So the sequence of primes above 3 is really two sequences that are the molds produced by the results of the calculations

            (6n + 1) * (6m - 1)

            (6n - 1) * (6bm + 1)

or

            (6n– 1) * (6m - 1)

            (6n + 1) * (6m + 1)

To reiterate, the series of prime numbers above 3 is really a combination of two sequences and each sequence could be examined separately. This blog examines the unprime composites more closely.

The patterns of the unprime composites

If we look at the results of calculating (6n+1) * (6m-1) for ascending values of n and m it becomes apparent that there are often two or more combinations that produce the same composite result. I refer to these as groups. For example 245 belongs to group 2 since there are two possible combinations. 

            245 = (6*1 - 1) * (6*8 +1) = 5 * 49

            245 = (6*1 + 1) * (6*6 –1) = 7 * 35            

With the help of computers, I calculated the unprimes up to 1 million. Table 1 shows the counts for ‘minus unprimes’ by group.

The X axis shows numbers up to 1 million in groups of 10,000. Within each group of 10,000 there can be most commonly 1666 or occasionally 1667 possibilities for (6n - 1) numbers. The X axis shows the count of the combinations out of the 1666 total combinations for each 10,000.

For example, from 0 - 10,000 there are 

• 616 prime numbers (blue)
• 743 unprimes such as 35 = 5 x 7 with only one set of factors ( red - the top line)
• 97 unprimes such as   with only two combinations of factors (yellow)
• 183 unprimes such as   with only three combinations (green)
• 23 unprimes with only five combinations (blue) 
• 3 unprimes with only seven combinations (brown)

These add up to 1665. In addition there is one unprime with only four combinations. I don't show it on the graph because it doesn't show up too well.

As expected, the graph for the counts of prime numbers shows the logarithmic curve. What is interesting is the logarithmic curve also applies to the combinations as well. For example, the green curve for the three combinations is a mirror image of the curve for primes.

In addition, some combinations increase and others decrease.

Let's extend the graph out to 10 million.

Now the pattern really becomes obvious. There is something going on here. Not shown on the graph are the extra curves for the additional groups with 4, 6, 8, 9, 11, 13, 14, 15, 17, 19 and 23 combinations, The counts for these groups are relatively small, but they do exist. They all appear to have this 'powers of x' shape.

How about the 'plus unprimes'.

It's not a surprise that the shape of the curves is very similar to the 'minus unprimes' curves. But they are not exactly the same. There is some asymmetry, particularly with the number of additional groups, but also with where these groups start.

                  Minus                                       Plus

group	start		group	start
0	5		0	7
1	35		1	25
2	245		2	175
3	455		3	385
4	6875		4	1225
5	1925		5	2275
6	171875		6	109375
7	6545		7	5005
8	13475		8	15925
9	48125		9	56875
			10	765625
11	25025		11	32725
			12	1500625
13	1203125		13	148225
14	336875		14	398125
15	85085		15	95095
17	175175		17	229075
19	625625		19	818125
20	8421875		20	9953125
			22	3705625
23	475475		23	425425
26	2277275		26	1926925
29	4379375		29	5726875
31	1616615		31	1956955
35	3328325		35	2977975
			39	10635625
47	9784775		47	8083075

The most important concepts I would like you to get out of all of the preceding are:

1 - There are two sequences, the plus and the minus. Although they are similar, they are different.

2 - Prime numbers are just one subset of the plus and minus sequences. The group with 47 combinations is just as important as the group with just one combination, or the group with zero combinations, the primes.

3 - When calculating the gap between primes, it is probably more meaningful to calculate the gap between primes from either one group or the other. The same with the distribution of primes.

If you haven't decided I am an absolute ratbag by now, you might look at the blog entry for the Goldbach Conjecture. There is interesting stuff there.

I also have some speculations about the Riemann hypothesis.

Twin primes

Before you read this blog, you need to read this blog about prime numbers belonging to one of two sequences, 6n + 1 and 6n - 1.

Let's look at Euclid's famous proof about the number of primes being infinite. The Wikipedia article is a little obtuse so take at look at this simpler version. It becomes obvious that Euclid's proof applies only to the 6n + 1 series.

In more detail, the Euclid proof of multiplying primes together, including the 2 and 3, and then adding 1 always creates a result belonging to the 6n+1 series, prime or otherwise.

Multiplying any number of primes above 3 by 2 and then by 3 and then adding 1 invariably produces a 6n+1 result, prime or otherwise.

Therefore Euclid’s proof applies only to the primes belonging to the 6n+1 series. Are there an infinite number of 6n-1 primes? Could we change Euclid’s method to subtract 1 instead of adding 1 to include those 6n-1 primes? As far as I can determine, the -1 proof works just as well as the +1 proof.

We we go back to the alternate explanation of Euclid's method and in step 2 substitute -1 for +1.

If this is so, we then have another way of looking at the twin prime problem. Essentially for every possible twin prime, there are four possibilities

1.      6n-1 prime                   6n+1 prime

2.      6n-1 prime                   6n+1 composite

3.      6n-1 composite            6n+1 prime

4.      6n-1 composite            6n+1 composite

Now we can look at the problem in a different light, essentially like looking at a pair of scissors with two infinitely long blades instead of a sword with one infinitely long blade. The traditional approach presumes that something special happens for a twin prime to exist at higher values. This alternate approach requires nothing special for twin primes to exist at higher values and would actually require some pattern to perpetually disallow twin primes at very high values.

If we can say that Euclid’s proof applies to both the 6n-1 as well as the 6n+1 series, then unless there is a recurring pattern to the distribution of primes, there is nothing to prevent a twin prime from existing at any stage.

Paradoxically, it is the lack of the so far unobserved recurring pattern for prime numbers that allows the twin primes to exist.

The other point to consider is that twin primes are inherently useless. They are no more significant than primes separated by 4, or a group 1 composite separated from a group 5 composite. Are there an infinite number of group 1 composites separated by 2 from a group 5 composites? Such questions all just as meaningless.

Goldbach Conjecture

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.

1. (6n-1) + (6m-1)    = 6n + 6m -2        
2. 3 + 6n+1              = 6n + 4
3. (6n-1) + (6m+1)   = 6n + 6m
4. (6n+1) + (6m-1)   = 6n + 6m
5. (6n+1)  + (6m+1) = 6n + 6m +2
6. 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

As you can see there are three groups, 6n -2 (minus) , 6n (even) , 6n +2 (plus).

Once you understand this, the Goldbach Comet graphs start to make a bit more sense.

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.

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.

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.

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.