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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Integer screens


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)

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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 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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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 the words “random” and “mysterious”. There is no doubt their distribution has been difficult to explain but just because something is “complex” does not make it random or unexplainble . 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.


Lines of Symmetry

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.

The first couple of lines of “symmetry” are well known. The first 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 next line of symmetry 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 smaller primes are discarded first. This diagram reflects a number of features of the primes which are obvious in retrospect:

i) there are only three types of pair of prime numbers 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),

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





The lines of symmetry are not immediately obvious if one only has the set of prime numbers to start from. It is necessary to start from the Sieve of Eratosthenes in order to obtain the necessary groupings (to be described later).

The third line of symmetry 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. Some of these bases aren’t prime …..121, 143, 169, 187, 209, 221…but are predictable. 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:

[to be repaired and continued after someone hacked in here]

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