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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Introduction


Prime Number distribution has been a source of fascination to both professional and amateur mathematicians for many years. The distribution is easy to explain, and conceptually easy to understand, once explained, but almost impossible to see if one looks just at the prime numbers themselves.

I believe that considering prime numbers as fundamental may have stood in the way of understand prime number distribution as their distribution is the result of composite number patterns, and the interference of these simple patterns that leads to the relative complexity of patterns in the primes.

The picture above of the two rows of trees in an avenue is not by chance. If the row of trees at one prime spacing combines with the row of trees at a different prime spacing one gets a combined pattern. Primes are just the left overs after all composite patterns have been taken into account.

Imagine two waves – one of wavelength 2 and one of wavelength 3; consider their interference pattern after 9 (for <9 see later). A familiar pattern emerges: at intervals of 6 three integers are multiples of 2 and two of 3; the resultant left overs (or potential primes) occur in pairs: thus 11,13 followed by 17,19 etc. The occurrence of pairs of primes are one of the noteworthy features of the prime series, and they occur simply as a result of first two primes. As factors of higher primes kick in (see later) the numbers of pairs declines. The question are their more potential pairs than there are multiples of higher primes (in which case pairs of primes continue indefinitely) or are there more multiples of higher primes than potential pairs (in which case pairs of primes disappear, but at what point?)

So what are the steps in understanding? Firstly, whilst primes may be the building blocks for integers, their pattern is determined by the composites. Prime distribution is determined not by one simple pattern, but a changing pattern that becomes more complex as integer number increases and in an infinite way. Secondly, the overall pattern is a composite of what are relatively simple patterns, each produced by a single prime factor. Thirdly, the composite patterns come into play at different starting points.

That’s about it. Once you have these concepts and the details (see below), prime number distribution is easy to explain……however, once you realise that the “simple” patterns become very long, very quickly you will see why visualisation of any pattern is simply beyond the capacity of the simple human brain.

You can also explain why repeatable patterns do occur in the primes, and where they come from!

Thus the overall pattern between 4 and 9 is not the same as it is between 9 and 25.

Also smaller prime factors influence the arrangement of the pattern of larger prime numbers.

Thus by integer 25 the component pattern that joins the overall prime pattern repeats itself at a length of 30 (2x3x5); however, the repeating patern only starts at 35 (not 25), whilst at integer 49 another component joins with a repetition stretching to 210 (2x3x5x7) but the repeating pattern only starts at 77 (=49+4*7). By the time you get to 121 you’re looking at a further pattern being added with a repeating pattern of length 2,310.

I hope these short paragraphs give you a little more insight into how prime numbers are distributed. Feel free to rate any pages you find useful, or comment.

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Prime Number Symmetry


Most texts on Prime Numbers (and even a professor of Maths at Oxford University?) don’t seem to explain the patterns and order in the primes and instead prefer to use words such as “random”, “unruly or “mysterious”. There is no doubt their distribution has been difficult to explain but just because something is complex does not make it random or unexplainable. Let’s be clear prime numbers are not random…..but even mathematicians are surprised when they find apparently non-random patterns:

https://www.nature.com/news/peculiar-pattern-found-in-random-prime-numbers-1.19550

https://www.newscientist.com/article/2080613-mathematicians-shocked-to-find-pattern-in-random-prime-numbers/

https://arxiv.org/pdf/1603.03720.pdf

I hope you find the diagrams below useful in understanding prime distribution and for predicting where primes can occur and where they definitely won’t. Comments welcome as I am learning all the time.

Symmetry is not given much attention when considering prime number distribution – hardly surprising as prime numbers don’t occur symmetrically! However if one looks at the positions that prime numbers occur it is clear that they occur only in certain locations, and don’t occur in other locations. In addition, the locations where primes do occur can be mapped using lines of symmetry.

As a first step, if you have some doubt about this calculate p#/2 and then look at the differences between the primes and p#/2 either side of p#/2. Do that for each p#/2 – yes e p#/2 gets big very quickly so representing the situation on a piece of paper or computer screen for larger p# is not practical.

As ever prime patterns are not simple and a number of anomalies appear: not all primes fit in but you have to persevere and the two diagrams below should assist you.

In general terms for each Pi   :

  1. The initial array (or helix) starts with Pi+1  (i.e. the next prime number)
  2. Calculate Pi-1 #   (yes, that’s the previous prime in case your eyes are as bad as mine)
  3. Layout the prime numbers from Pi+1
  4. to   Pi+1  + Pi-1 #
  5. Then calculate all composite numbers for Pi  with larger prime numbers Pi+1 (but not smaller ones) ……as far as Pi+1  + Pi-1 #
  6. Combining the primes with the calculated composites then gives you the base line for each array/helix.

The first couple of lines of “symmetry” are trivial and associated with prime numbers 2 and 3. The third – associated with prime number 5 – reflects the fact that most (not 2 or 3) prime numbers can be expressed in the form 6n+1 and 6n-1, but with the disappointing aside that not all such numbers are prime. Within each strand of primes they are distributed at intervals of 3# = 6. As with further lines of symmetry, the smallest prime numbers don’t fit in, but in a predictable way – the patterns are stepped, becoming more complex the larger the integer. Thus the symmetrical patterns do not all start at 2, but at increasing primes. This is fundamental to understanding how primes are distributed.

Thus, putting 2 and 3 to one side, the prime numbers can be considered to be either left handed with a formula of 5 + n.3# (5,11,17,23…) or right handed with a formula of 7 + n.3# (7,13,19,31….). In theory at least the line of symmetry starts at 3 (or 3#/2).

The fourth line of symmetry derives from prime number 7 and starts at 5#/2 = 15 AND the pattern it explains starts at a larger prime, 11. The primes occur at intervals of multiples of 5#=30 from 8 bases:


with ALL prime numbers distributed at equal intervals about the line 15-45-75-105…. provided that primes  <= Pi  are not included. This diagram reflects a number of features of the primes which are obvious in retrospect:

i) there are only three types of prime number pairs   1:3, 7:9 and 9:1.

ii) the 1:3 pairs match the locations of the 7:9 pairs about the line of symmetry; the 1:9 pairs are equidistant from the line of symmetry

iii) there are two lines of primes that never form pairs (the +8 and -8 lines), so we can speculate that prime pairs contain more primes that end in 1 and 9 and fewer that end in 3 and 7. ….that in some respects assumes that each of these 8 lines contain an equal frequency of primes….which is also an interesting thing to consider further.

iv) there are zones of 6 integers on either side of the line of symmetry in which primes never occur (i.e. between the +8 and +14 lines) between primes ending in _1 and _7 on the one hand and those ending between _3 and _9.

In consequence it can be conjectured that there are an infinite set of similar lines of symmetry in the prime numbers each starting at n#/2; thus

105…315

1155…3465

15015…45045

etc

The lines of symmetry are not immediately obvious if one only has the set of prime numbers to start from. If you follow the instructions above you should be able to construct the base line for each helix.

The fifth line of symmetry – derived from the fifth prime number 11 – starts at 105; primes occur at intervals of 7#=210. As shown it is not quite symmetrical and two lines from the bottom – starting 211 and 221 – have to be brought up to the top, but staggered, to provide the complete pattern. At first sight it doesn’t look symmetrical but once transferred it’s OK.

Some of these bases aren’t prime …..121, 143, 169, 187, 209, 221…but are predictable. How? Well they represent 11×11, 11×13, 13×13 (i.e.multiples of all primes 11 and greater). Notice again the pairing of prime pairs about the line of symmetry, and the absence of primes in some zones of length 6, and some areas with no pairs:

Jon Heuch

Bringing the last two lines up to the top gives the complete picture:

PF11 helix

So looking at the pairs again:

There are 48 lines:

7 lines of 3 and 7 with no chance of pairing (14 lines)

2 lines of 1 and 9 with no pairs (4 lines)

So pairs only occur on 48 – 14 – 4 lines: 30 lines – 7 pairs of lines on each side of the line of symmetry and one pair that straddles – equidistant starting 209-211. There are 5 pairs of each 1:3, 9:1 and 7:9……all very orderly!

 

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The balance of the primes – conjecture


Primes align.

This first becomes apparent with the bases 5 and 7; all subsequent primes can be stated as multiples of 3#, adding a base of either 5 or 7.

The next alignment occurs using the eight bases 11, 13, 17, 19, 23, 29, 31, 37. All subsequent primes can be stated as multiples of 5#, adding one of these 8 bases.

Subsequent alignments are based on larger numbers of bases as each p# is considered. At the next the number of bases becomes 48 and the interval 7# or 210.

The question arises are primes balanced between bases? Are there as many primes based on 5 as 7? Over short lengths of primes this is not the case; but over longer runs there seems to be some balance.

Similarly, the 8 bases listed above, there seems to be some sort of balance, but it is anticipated there will be fluctuations but possibly no long term bias?

A nice project for some student?

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The interdependence of Primes


David Wells’ book on Prime Numbers notes the parallels between lucky numbers and primes, suggesting the role of the sieve (p147/8 under “random” primes). Imagine what the effect would be if one prime was no longer considered to be prime. When one realises the interdependence of the primes some of the relationships can be seen:

The pattern in the primes changes at each p squared, but the distance between each p squared is not independent – it is determined by smaller prime numbers: thus the pattern of the primes is fixed between 9 and 16; the length of this pattern can be determined by the simple formula 2.PF.a+a squared where PF is the square root of the lower boundary (a prime) and a is the difference between this prime and the next smallest prime (i.e. pn – pn-1). Thus if we look at each largest gap in the prime numbers we find that we can predict a relatively long section of prime numbers with the same pattern i.e. the gap between 113 and 127 of 14 causes the pattern between 12,769 and 16,129 to be uninterrupted. Of course noone has ever noticed this as the pattern starting at 12,769 is a mere 113# long (i.e. big!) so the run up to 16,129 presents just a fraction of the pattern before the new pattern starts at 16,129.

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