Milestones in the primes – calculating the n in pn


So, in the two previous blogs on composite and secondary arrays I have shown how we can lay out prime numbers in an infinite series of arrays, with gaps filled by specific composites made up of pn x pn and greater but not involving an composites having a smaller denominator.

These arrays appear to be a little disappointing if we are only interested in prime numbers.

If I could only get a table into a jpg file format I would post it here! Inserting a table seems impossible to keep the format in wordpress! In the mean time it doesn’t take much to leap from the infinite series of arrays provided to be able to show where specific numbered primes will i.e. in the table below

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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Secondary arrays


This element needs to be read after that titled “Composite arrays“.

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 Pn 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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Composite arrays


This blog may seem to be all about composite numbers. You might well ask “What has this to do with prime numbers?”. It’s a good question, but you need to read on through the next couple of blogs to see a startling result : when we talk about n of Pn   what do we know?

For every Pn prime number there is an associated array of composite numbers that consist of Pn x Pn and greater composites but no composites with denominator of Pn-1 or less.

As you will see below the sequence of composites are well defined, but their repetitive positions are less well known – one can think of the arrays as spirals rather than just a list.

Each of the arrays for n = 1 and n=2 i.e. for prime numbers 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 n=3 (i.e. prime number 5) does the array start to have some life with two lines of integers, but it takes examination of n=4 and n=5 before some interesting things start to emerge. Each column of the array for n=4 has 8 lines of integers and that for n=5 has 48.

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

  1. there are  (Pn-1-1)# elements in each column
  2. the columns are  Pn# apart
  3. each array has a line of symmetry at    Pn#/2

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

Pn        No of elements in 1st column  Gap between columns   Line of symmetry

2                1                                                2
3                1                                                6
5                2                                               30                                         15
7                8                                               210                                       105
11              48                                             2310                                     1155
13             480                                           30030                                  15015
______(Pn-1  – 1)#                                  p#                                          p#/2

OEIS      A005867                                A002110                                A070826

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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Counting the prime pairs


Is there an infinite number of prime pairs?

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?

 

 

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Prime and Composite


Integers are either Prime or Composite. Think of the composites as a wood block with pattern encrypted and the Primes as the resulting print. However, you can stare at a block of wood for a long timer without seeing the pattern that results.

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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Prime or Composite?


Is an integer Prime or Composite? Is there any proof an integer being Prime, other than it not being Composite i.e. it has no prime factors? Lists of potentially prime numbers or pseudo primes http://en.wikipedia.org/wiki/Pseudoprime have been identified using a variety of methods. The method below has possibly not been identified before (?)

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

Pn = Pi + nPj#

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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p+np# – a formula for the primes?


Take virtually any prime number as a start, and add any multiple of the next, smaller, prime number and you will create a set of integers that contains every prime…….provided one takes into account the structure of the primes as outlined on this blog. So 5 has to be considered with 7; add 3# or 6 to either 5 or 7 and one creates a set of integers containing every prime.

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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