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 P_{i :}

- The initial array (or helix) starts with P
_{i+1 }(i.e. the next prime number)
- Calculate P
_{i-1 }# (yes, that’s the previous prime in case your eyes are as bad as mine)
- Layout the prime numbers from P
_{i+1 }
- to P
_{i+1 } + P_{i-1 }#
- Then calculate all composite numbers for P
_{i }with larger prime numbers P_{i+1 }(but not smaller ones) ……as far as P_{i+1 } + P_{i-1 }#
- 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 <= P_{i} 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:

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

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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