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.