This blog may seem to be all about composite numbers. You might well ask “What has this to do with prime numbers?”. It’s a good question, but you need to read on through the next couple of blogs to see a startling result : when we talk about n of P_{n } what do we know?

For every Pn prime number there is an associated array of composite numbers that consist of Pn x Pn and greater composites but no composites with denominator of Pn-1 or less.

As you will see below the sequence of composites are well defined, but their repetitive positions are less well known – one can think of the arrays as spirals rather than just a list.

Each of the arrays for n = 1 and n=2 i.e. for prime numbers 2 and 3 are trivial in that they each consist of a single line of integers (4,6,8,10 etc and 9,15,21,27 etc). Only with n=3 (i.e. prime number 5) does the array start to have some life with two lines of integers, but it takes examination of n=4 and n=5 before some interesting things start to emerge. Each column of the array for n=4 has 8 lines of integers and that for n=5 has 48.

There are three dimensions I can give for each of these arrays:

- there are (P
_{n-1}-1)# elements in each column
- the columns are P
_{n}# apart
- each array has a line of symmetry at P
_{n}#/2

So here is a starting table (sorry for the limitations of posting a table in wordpress):

Pn No of elements in 1st column Gap between columns Line of symmetry

2 1 2

3 1 6

5 2 30 15

7 8 210 105

11 48 2310 1155

13 480 30030 15015

______(Pn-1 – 1)# p# p#/2

OEIS A005867 A002110 A070826

For those who have explored the prime numbers the above framework may appear familiar. However, arrays of composite numbers, by themselves tell us very little about the primes. You have to move onto the secondary arrays to get an inkling on what is going on…..see “**Secondary arrays**“.

But before moving on we need to examine the composite integers that sit on the above arrays or frameworks. As above it is useful to refer to the Online Encylopedia of Integer Sequences (OEIS) rather than listing long lists of integers: however, there are a number of differences with regards to starting points. I haven’t been too careful on this front so I expect some quality control might be useful.

So, for the composite integers that “sit” on the above frameworks:

OEIS sequence A083140 contains a pretty good summary of the early sequences, but of course the arrays are not acknowledged. The following OEIS sequences make up A083140 but in this context you have to leave off both the first prime number and the second term of the prime squared:

`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)`

Thus the repetitive array for prime number 7 starts with 77 and the second column of the array starts with 77 + 210 = 287. Thus the composite array as a whole reads:

77 287 497 707 917 1127 1337 1547

91 301 511 721 931 1141 1351 1561

119 329 539 749 959 1169 1379 1589

133 343 553 763 973 1183 1393 1603

161 371 581 791 1001 1211 1421 1631

203 413 623 833 1043 1253 1463 1673

217 427 637 847 1057 1267 1477 1687

259 469 679 889 1099 1309 1519 1729

As you can see there are 8 rows of integers, and each item in each row is spaced at 210; however, the “line of symmetry” at 105 is not in the middle. One has to rearrange the array to “see” it – two lines need to be brought to the top of the array and offset:

____217 427 637 847 1057 1267 1477 1687

____259 469 679 889 1099 1309 1519 1729

77 287 497 707 917 1127 1337 1547

91 301 511 721 931 1141 1351 1561

————–line of symmetry————————

119 329 539 749 959 1169 1379 1589

133 343 553 763 973 1183 1393 1603

161 371 581 791 1001 1211 1421 1631

203 413 623 833 1043 1253 1463 1673

(Offsetting or steps is something one gets used to with primes, but we’re not on the primes yet!)

So now if one measures the distance of each of the eight rows from a line starting with p#/2 you will see the rows are spaced at equal distances from the mid-line:

-98 (the distance between 217 and 315)

-56 (the distance between 259 and 315)

-28

-14

+14 (=2 x 7)

+28 (= 4 x 7)

+56 (=8 x 7)

+98 (= 14 x 7)

So, an array of equal spaced composite numbers is not too exciting is it? Just keep following – it gets much more interesting: see “**Secondary arrays**”