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……

Consider posting your unique sequences to the On-Line Encyclopedia of Integer Sequences. oeis.org

See this link: https://oeis.org/history?seq=A225776 and click through the “Older changes” for the exposition of a sieve based on digital roots. Very similar to some of the stuff you and Croft get into on this blog.

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Thanks. Yes I think I am following long behind others in this area so am not expecting to have found anything new. Just wish others would explain things more clearly!

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