Firstly the composite helices and thus the prime helices each have a “width”. The width of the 6n+/-1 helix associated with prime factor 5 is 2. The width of the next helix associated with prime factor 7 is 8. The next is 48 and the conjecture is that this continues:

480

5760

etc

Secondly, considering the balance of primes to composites as each prime factor is used to calculate which are composite and which are potentially prime we see a simple pattern:

Each block is p_{i}# integers long,

Each block has (p_{i } – 1) # (potential) primes & (p_{i-1 }-1)# composites

Thus the ratio of

primes to all integers: (p_{i } – 1) # / p_{i}#

primes to composites (p_{i } – 1) # / (p_{i-1 }-1) # = p_{i } – 1

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A) Integers are either composite or prime; think of their relationship rather like a block print and the resulting print: they both have patterns on them but it is the resulting print that we all look at and admire; we look at the block and can make no sense of it.

Humans have been looking at the prime “print” for a long time and made little sense of it; my answer is to look at the composites – it is quite eye-opening!

B) The prime pattern is an interference pattern between waves; you can find references to music and diagrams of waves but few appear to have taken if far. This is another line of attach worth examining but only after A)

In summary, what have I to report:

- The prime “pattern” expressed by prime numbers is in fact a composite of one to many individual patterns, each arising from a string of composites…..derived from each prime number – there is one relatively “simple” pattern for each prime number – OK not so simple and only the early few are short.
- The prime “pattern” is of a stepped nature so it starts simple (2,3….how more simple do you want it…..5….7?) and gets more complex with each step. The steps are at P
_{i}^{2}. - Each “pattern” of composites that contribute to the overall “pattern” are of length P
_{i}#. - Each “pattern” of composites does not come into effect at P
_{i}^{2}. Each starts at P_{i}^{2}+ a variable multiple of P_{i}. - There is a drawbridge effect so that the overall pattern “progresses” and the distribution of the smaller primes are too simple to be included in the pattern for larger primes. This is one of the main reasons why virtually every rule for the primes we can define has an exception…..e.g. all primes are odd EXCEPT 2!

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Barriers to doing this:

i) whilst the general trend of the primes becoming less frequent with increasing n is well known the frequency does not decrease uniformly; the terms appear to pulse with both gaps and areas of higher frequency.

ii) the well known twin primes (but no triplets (see Counting the Twins)

iii) the well known gaps – with evidence of increasing gaps with increasing n

Primes have been described as random, unruly, baffling reflecting the frustration of the observer; there has also been commentary on harmony, music and other references to spirals, waves and the like. Just what is going on? Explainable or inexplicable?

In addition there are several well known observations about the primes:

Almost all rules/observations require an exception e.g.

- all primes are odd, except 2. So even the simplest description has an exception!
- all primes are separate, except 2 & 3.

Which leads onto a more general problem:

- rules for smaller primes may not apply to all (i.e. larger) primes
- rules that appear to apply require exceptions of smaller primes

As you will read in this blog once one understands the nature of the prime number problem and the inherent complexity, things become clearer, at least conceptually. Understanding that structure gives me some confidence to predict that there will never be a practical prime formula and since I can say why. However, there does seem to be a way of counting primes – but not every prime – and thus making some sense of things.

The startling thing is that the analysis is relatively simple…….

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This blog has shown how it is possible to lay out specific composite numbers in a series of arrays or helices. Each array/helix is related to a specific prime number and the composite numbers in each array/helix are prime multiples of that specific prime number. The composite array/helix can be translated into a secondary array/helix by dividing each composite by the specific prime number. The secondary array is rich in prime numbers and the composites that fill the gaps in the secondary array are all multiples of the specific prime number and all larger prime numbers (i.e. multiples of smaller primes are excluded). The important thing to bear in mind is that ALL primes larger than the specific prime number are included AND that the primes are arranged in an array. With a little careful transfer of “lines” (i.e. rows or columns depending on how you have arranged them) a “line of symmetry” at p#/2 runs through the middle of each secondary array/helix.

The next thing to bear in mind is the number of positions in the each array/helix. This is crucial to counting primes and identifying the *i *^{th }term. The number is the same for both primary (i.e. composite) and secondary (i.e. prime rich) array.

So the array associated with prime number 5 has a mere 2 positions. As we have seen this leads to the well known 6n+/-1 formula.

The array associated with prime number 7 has 8 positions. Due to the absence of composites in the secondary array (i.e. the first composite appearing is 49) we can count the primes:

The array – associated with prime number 7, the fourth prime number – starts with 11, which is the fifth prime number (i.e. +1 from 4 to 5). We know there are 8 lines in the array so the first prime number in the second line will be the 13th prime number. Based on the more advanced/larger arrays we cannot predict exactly what the 13th prime number will be but we do know that the NEXT prime number will be the 13th. In this case the next prime is + 5#=30 i.e. 11 + 30 = 41.

Moving on we can see that adding p_{i }to p_{i-1} # gives a good way of counting the primes as we know the number of primes separating p_{i }from p_{i }+ p_{i-1} # .

Looking at the first prime in the second row of each array/helix:

13th prime is 41

48th prime is 223

345th prime is 2327……except that 2327 is not prime: 2333 is the 345th prime so you will need to see the table to see what it provides

3,250 etc….

42,333

646,031

12,283,533

300,369,798

8,028,643,012

259,488,750,746

9,414,916,809,097

Meanwhile the series 13, 48, 345, 3250, 42333 is not registered on OEIS so maybe no one has got here yet?

Meanwhile I have found out that Microsoft Excel starts to be of little use if you need more than around 15 significant digits so this is as far as I can go:

The 9,414,916,809,097th prime number is the first prime number after 304,250,263,527,257.

I can make a stab at both the

362,597,750,396,746th prime number

and the 1,539,772,852,781,310th prime number

but excel being excel and I have also reached the limit of primes.utm.edu

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A secondary array of integers (a mix of primes and composites) emerges by dividing the associated composite array by Pn i.e. by the prime number “associated” with that composite array. Remember, the array is still in place but this time, whilst there are still composites in the array, a remarkable thing happens – all the prime numbers P_{n }and greater emerge. On contemplation, a mix of primes and composites may not seem very enlightening. However, it appears that the composites are “disappearing” faster than the primes.

Let’s look at the secondary array associated with prime number 5:

The composite array consists of

25

55 85 115 145

65 95 125 155

The resulting secondary array consists of two rows with a gap of 6 between the columns

11 17 23 29

13 19 25 31

This is recognizable by the well known 6n+1 and 6n-1 algorithm that contains all primes. It appears to be less well known that there is an infinite number of such arrays (one for each prime) arising from this process.

Let’s look at the secondary array that arises from prime number 7. The composite array is shown under the composite array blog so if we divide each term by 7 we are left with

11 41 71 101 131 161 191 221

13 43 73 103 133 163 193 223

17 47 77 107 137 167 197 227

19 49 79 109 139 169 199 229

23 53 83 113 143 173 203 233

29 59 89 119 149 179 209 239

31 61 91 121 151 181 211 241

37 67 97 127 157 187 217 247

So, we still have eight rows but now the columns/terms are separated by 210/7 i.e. 30. The first column contains only prime numbers but composites creep in at greater frequencies as you move left to right (not surprisingly). However, the composites are not just any old composites – they are an increasingly select bunch that fill the gaps between primes, which, remember, are in an array – those naughty old primes, often accused of being “random” are in a set matrix at regular intervals. And remember, there is an infinite set of arrays, one for each prime number!

So, let’s do the same as before to see the “line of symmetry: move two lines from bottom to top and offset:

___31 61 91 121 151 181 211 241

___37 67 97 127 157 187 217 247

11 41 71 101 131 161 191 221

13 43 73 103 133 163 193 223

——-line of symmetry at pn-1# i.e. 15——

17 47 77 107 137 167 197 227

19 49 79 109 139 169 199 229

23 53 83 113 143 173 203 233

29 59 89 119 149 179 209 239

Having divided the composite array by 7 the lines are now separated from the line of symmetry by

+/-14

+/-8

+/-4

+/-2

So an infinite series of arrays with a mix of primes and composites – mysterious perhaps, but not very exciting. Hold one….let’s see what come next: “**Milestones in the primes – calculating the n in pn**”

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A practical difficulty kicks in quickly – well two difficulties actually:

- It’s not as simple as counting each multiple of 2,3,5,7,11 etc..separately (i.e. each prime number or factor) as all composites (other than squares) are the result of multiplying two different primes so can be expressed more than once; thus 6 = 2 x 3 =3 x 2 and so could be counted as a multiple of 2 and a multiple of 3. As another example 60 = 2 x 2 x 3 x 5 could be counted as a multiple of 2, 3 and 5. An adjustment is required.
- The first factor leads to a second related difficulty in that no composites need to be counted against any prime factor P
_{i }where c < P_{i}^{2 }. Thus for prime factor 5 the first composite not already accounted for by prime factors 2 and 3 is 25. The next is 35. Smaller composites such as 10, 15, 20 and 30 are also multiples of 2 or 3 or both. - Once the second factor is taken into consideration we don’t need to worry about adjusting any count for primes themselves.

So let’s make an adjustment for these two difficulties, assuming we start with the smallest prime factor 2 and work upwards:

The number of composites from 2 starts with 4 and continues with every larger even number: as a series these can be expressed as 2n+2 (i.e. where n=1 the term is 4) or counting from the integers as a whole n = I/2 – 1 or (I-2)/2

i.e.

I=4 n=1

I=6 n=2

Thus is we wish to count the number of composites of 2 less than an integer I, we can use I/2 – 1.

We can then count the number of composites from prime factor 3. These start with 9 and continue with every odd multiple of three (the even ones were all accounted for as multiples of 2. The formula 6n+3 applies (i.e. where n=1 the term is 9).

To count the multiples based on integer I we use the formula (I-3)/6.

Combining the count for prime factors 2 and 3 gives a formula of 2(I-3)/3 where the formula produces a whole integer.

Can we continue with this approach? Trouble is, we cannot (easily) as the composites for prime factor 5 (and larger factors) are not uniformly distributed 25,35, 55,65 etc. For easy formulas we need to consider two separate formulas; for prime factors 7 and above we need multiple formulas….the number of which amazingly relate to Euler (see elsewhere).

At this point we need to take a closer examination of the unique composites associated with each prime factor. In the paragraph above the start of the sequence associated with prime factor are listed. As will be seen they occur in pairs and are 30 apart; as will be seen elsewhere these define the well known 6n+1/-1 formula (divide them by 5 if you cannot wait, but one of the tricks is to work out where exactly the series starts).

Things start to get a little “clearer” (maybe complex is a better word initially, but clearer can be used in retrospect) when we look at the unique composites associated with prime factor 7.

With no surprise they start at 49 but then continue in what at first appears a little random.

49, 77, 91, 119, 133, 161, 203, 217, 287, 301, 329, 343, 371, 413, 427

It’s only when you arrange them that you start to see a bit of structure

49,

77, 91, 119, 133, 161, 203, 217, 259

287, 301, 329, 343, 371, 413, 427, 469

putting 49 to one side you can see that the positions of the composites are in rows of 8 with a repeat at intervals of 210. You can keep going for as long as you want; after all the pattern is a result of an interaction of the four prime factors – 2, 3, 5 and 7 – nothing else so this is pretty simple.

If you are interested in the Online Sequence of Integers https://oeis.org/ this sequence is labelled** A063163** and defined as “Composite numbers which in base 7 contain their largest proper factor as a substring”.

We can do this analysis for each subsequent prime factor.

We find that for every prime number p_{i }there is an associated array of composite numbers that consist of the composites derived from p_{i }and larger prime factors but does not include any composites with a prime factor of p_{i }or less.

Each sequence of composites are well defined and can be considered to be arranged in a helix but the start of each subsequent helix is off-set by one prime factor.

The first two “helices” (for prime factors 2 and 3) are trivial in that they each consist of a single line of integers (4,6,8,10 etc and 9,15,21,27 etc). Only with that for prime factor 5 does the helix start to show us some of its features with two lines of integers, but it takes examination of the fourth and fifth before a fuller picture emerges…..and if you have got this far and wondered what this is going to tell you about prime number distribution, just be patient!

There are three dimensions I can give for each of these composite helices:

- there are (P
_{i-1 }– 1)# elements in each turn of the helix - the strands of the helix are p
_{i}# apart - each helix has a line of symmetry at p
_{i }#/2

So here is a starting table (sorry for the limitations of posting a table in wordpress):

p_{i} No of elements in 1st column Gap between columns Line of symmetry

3 1 6

5 2 30 15

7 8 210 105

11 48 2310 1155

13 480 30030 15015

______(P_{i-1 }– 1)# p# p#/2

For those who have explored the prime numbers the above framework may appear familiar. However, arrays of composite numbers, by themselves tell us very little about the primes. You have to move onto the secondary arrays to get an inkling on what is going on…..see “**Secondary arrays**“.

But before moving on we need to examine the composite integers that sit on the above arrays or frameworks. As above it is useful to refer to the Online Encylopedia of Integer Sequences (OEIS) rather than listing long lists of integers: however, there are a number of differences with regards to starting points. I haven’t been too careful on this front so I expect some quality control might be useful.

So, for the composite integers that “sit” on the above frameworks:

OEIS sequence A083140 contains a pretty good summary of the early sequences, but of course the arrays are not acknowledged. The following OEIS sequences make up A083140 but in this context you have to leave off both the first prime number and the second term of the prime squared:

`2 4 6 8 10 12 14 16 18 20 22 24 .... (A005843)`

`3 9 15 21 27 33 39 45 51 57 63 69 .... (A016945)`

`5 25 35 55 65 85 95 115 125 145 155 175 .... (A084967)`

`7 49 77 91 119 133 161 203 217 259 287 301 .... (A084968)`

`11 121 143 187 209 253 319 341 407 451 473 517 .... (A084969)`

`13 169 221 247 299 377 403 481 533 559 611 689 .... (A084970)`

Thus the repetitive array for prime number 7 starts with 77 and the second column of the array starts with 77 + 210 = 287. Thus the composite array as a whole reads:

77 287 497 707 917 1127 1337 1547

91 301 511 721 931 1141 1351 1561

119 329 539 749 959 1169 1379 1589

133 343 553 763 973 1183 1393 1603

161 371 581 791 1001 1211 1421 1631

203 413 623 833 1043 1253 1463 1673

217 427 637 847 1057 1267 1477 1687

259 469 679 889 1099 1309 1519 1729

As you can see there are 8 rows of integers, and each item in each row is spaced at 210; however, the “line of symmetry” at 105 is not in the middle. One has to rearrange the array to “see” it – two lines need to be brought to the top of the array and offset:

____217 427 637 847 1057 1267 1477 1687

____259 469 679 889 1099 1309 1519 1729

77 287 497 707 917 1127 1337 1547

91 301 511 721 931 1141 1351 1561

————–line of symmetry————————

119 329 539 749 959 1169 1379 1589

133 343 553 763 973 1183 1393 1603

161 371 581 791 1001 1211 1421 1631

203 413 623 833 1043 1253 1463 1673

(Offsetting or steps is something one gets used to with primes, but we’re not on the primes yet!)

So now if one measures the distance of each of the eight rows from a line starting with p#/2 you will see the rows are spaced at equal distances from the mid-line:

-98 (the distance between 217 and 315)

-56 (the distance between 259 and 315)

-28

-14

+14 (=2 x 7)

+28 (= 4 x 7)

+56 (=8 x 7)

+98 (= 14 x 7)

So, an array of equal spaced composite numbers is not too exciting is it? Just keep following – it gets much more interesting: see “**Secondary arrays**”

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How frequent are prime pairs?

Why are their prime pairs?

Intriguing questions best examined from the composites: why do they exist for starters? Look at the role of prime factors 2 and 3. They lead to the first prime pair 5&7.

Look at the role of prime factors 2, 3 and 5. 41/43 and 59/61 show up, but 37/39 and 53/55 would be pairs without the effects of 13 and 11.

We can show the number of pairs per length of composite numbers thus:

It starts:

PF 2,3: 1 pair for each 5 composites (20% or 40% of composites are one of a pair)

PF 2,3,5: 4 pairs for 30 composites (13.3% or 26.6% of composites are one of a pair)

PF 2,3,5,7: 19 pairs in 210 composites (9.0% or 18.1% of composites are one of a pair)

and so on. Can we establish a formula and work out whether it has an asymmtote?

(p-1)#/p#

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Patterns in prime numbers are the result of patterns in the composites. Understanding the complexities of patterns in the composites gives us insights into patterns in the primes.

So the patterns in the composites are obvious aren’t they? So obvious that it’s not worth considering them?

The 2x table 2,4,6,8,10,12…..leads to the simple observation that all primes are odd.

The 3x table is equally uneventful 3,6,9,12,15………..Only half of these make an “imprint” on the primes as the 2x table has removed all even numbers. Thus 3,9,15,21,27,33….. uneventful, regular numbers.

The 5x table starts to show the issues: All even numbers and every 15 taken up by the 2x and 3x tables. The integers scored by the 5 x table are 5, 25,35, 55,65, 85,95….a regular beat with alternate gaps of 10 and 20 after the initial integer.

We see the pattern emerging of the prime number, the prime number squared followed by a regular repeating pattern.

The 7x table shows some of the further complexities:

Again p: 7 and and p squared: 49 start the process off. The list that follows appears random at first:

77 | 91 | 119 | 133 | 161 | 203 | 217 | 259 |

287 | 301 | 329 | 343 | 371 | 413 | 427 | 469 |

However, it becomes clear that the spacing of the second line mirrors that of the first, with the pattern repeating itself at p#, in this case 210 or 7 x 5 x 3 x 2.

28 | 14 | 28 | 14 | 28 | 42 | 14 | 42 |

The same process with each subsequent prime can be followed, with similar effects: the repetitive element is based on p#, whilst the spacing between are multiples of p.

The stepped nature of the composite number patterns, and their overlapping nature as a result of the long p# length patterns show us why examination of the prime numbers alone will give little away……

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Many primes can be written in the form, in multiple ways:

P_{n} = P_{i} + nP_{j}# |

Where i and j are related thus:

(to be continued)

Take any Integer and find the largest p# that is smaller than the Integer. Take the largest multiple of p# away from the Integer that leaves a positive Integer. Is the remainder prime or composite? If composite the likelihood is that the Integer is not prime (but this is not guaranteed). However do the same with the next smaller p#, & follow the same procedure – is the remainder prime or composite. Continue using smaller p#, if the series of remainders are prime seem to be chances of a prime. If the remainders are composite it’s not possible to suggest that the prime factors are, just that it is less likely to be prime

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Take 11, add 5# or 5x3x2=30, take the structure of the primes in that there are now 8 bases and a similar result obtains.

…..this is unfinished business, but thought it’s worth posting – a very simple formula – p+np# – throws up some interesting insights into prime distribution and complexity.

The page titled “prime number alignment” shows the outcome of this formula – numerous links between prime numbers! Mathematicians may not be surprised or intrigued but I thought it was at least interesting and worth noting.

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