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 P_{n }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**”