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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