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Author – Andrew J Frost 10/06/2020 REV TWO PnonP p7

OBSERVATION 7 Patterns:- Gaps between the Prime and Non-Prime Numbers in Set S_af

Interestingly and most importantly; following on from the integer “Raised By”, the Uplift pattern and then the “Last Two Ending Numbers” pattern (post -05); another pattern also fits in a similar way.

If, instead of just limiting the calculation of gaps between the prime numbers in the prime number series; the whole list of primes and non-primes is used, as in my special Set S_af, then a regular repeating pattern for gaps between all these numbers is evident.

The Gaps – Complementarity

Gaps in this series start with 1; that is the gap between 0 and 1. Then the repeating cycle starts as 6, 4, 2, 4, 2, 4, 6, 2. Without fail the sequence repeats thereafter throughout the entire number list Set S_af continually, as shown in the spreadsheet.

So from 1 to 7 the gap is 6, from 7 to 11 the gap is 4, from 11 to 13 the gap is 2, and from 13 to 17 the gap is 4 and so on.

See example from spreadsheet at the end of this post.

Gaps Spacing

The irregular spacing between primes is compensated for exactly by a similar but complementary irregular spacing between the special subset of odd-composite non-primes. The combination of these two subsets of numbers from then on forms a rigidly regular repeating pattern in Set S_af.

The starting numbers for the sequences, the first four rows, are a special case in response to the uplift of the four specific positive primes that occur below 12, when the initial integer in the list is 0. However due to the sifting out of the number 5, the uplift numbers shown start 1, 7, 11, as in column 3 below.

The primes are deeply linked into the non-prime odd-composite series. If one imagines the zip of a garment, one side of the zip is the series – ‘Set of the prime numbers’, then the other side is the ‘Set of the odd-composite numbers’.

The point being that a zip cannot function with only one half of itself.

Odd-Composite Factors

The odd-composite numbers are composed of factors which are the odd prime numbers themselves; only limited by being equal to or greater than 7.

The series does not vary, while proceeding upward along the number line, ad infinitum.

Regardless of the size of the numbers concerned or the actual size of the gaps between one prime and the next; these gaps are filled with repeating sequences of non-prime, odd-composite numbers.

This gap sequence of 8 numbers can be doubled to 16 and then rigidly matched against the “raised by” uplift series previously discussed in a previous post.

The fact that one Set of numbers (odd-composites as defined) – is exactly complimentary to another Set (prime numbers), I find more than fascinating.

Raised By Uplift	Ending Digit	Gaps Pattern
1	1	1
7	7	6
11	1	4
1	3	2
5	7	4
7	9	2
11	3	4
5	9	6
7	1	2
1	7	6
5	1	4
7	3	2
11	7	4
1	9	2
5	3	4
11	9	6
1	1	2
7	7	6

The table below shows the additional line including uplift 5. For reference only.

Starting Integer	Base times 12	Raised by Number	Ending Digit	Gap
0	0	1	1	1
0	0	5	5	4
0	0	7	7	2
0	0	11	1	4

–

Five Columns and Thirty-Two rows

The first four rows above are a special case in response to the uplift of the four specific positive primes that occur below 12, shown in column 3; ending digits – column 4; and gaps in column 5.

The integer ‘n’ and times 12 base columns are shown prior to these below, which is their logical position.

• (Note: the column numbers or positions might occur differently depending on the spreadsheet version.)

Doubling the Rows to Achieve the Overall Relationship

Below the table columns have to be doubled in length to 32 cells downward to accommodate a sequence match across all the rows. This pattern now repeats continuously every 32 rows and does not change as a subset of Set S_af on the number line.

• 1^st column shows the ending digit of the starting number integer ‘n’.

• 2^nd column is the ending digit of the base number x12.

• 3^rd column is the “raised by”, prime uplift, which raises the product of the x12 base number and the starting number. This produces either the next odd-composite non-prime or prime in the sequence.

• 4^th column shows the single ending digit of the prime or odd-composite non-prime produced.

• 5^th column is my gap sequence across the primes and odd-composite non-primes.

The table below shows the relationship between the various sequences of numbers. It is important to realise that none of these sequences vary except the first line of the ‘gap sequence’. They are immutably fitted together in only this one relationship.

Row No.	Starting Number	Base No. x Twelve	Raised By Uplift	Ending Number	Gap Sequence
1	0	0	1	1	1
2	0	0	7	7	6
3	0	0	11	1	4
4	1	2	1	3	2
5	1	2	5	7	4
6	1	2	7	9	2
7	1	2	11	3	4
8	2	4	5	9	6
9	2	4	7	1	2
10	3	6	1	7	6
11	3	6	5	1	4
12	3	6	7	3	2
13	3	6	11	7	4
14	4	8	1	9	2
15	4	8	5	3	4
16	4	8	11	9	6
17	5	0	1	1	2
18	5	0	7	7	6
19	5	0	11	1	4
20	6	2	1	3	2
21	6	2	5	7	4
22	6	2	7	9	2
23	6	2	11	3	4
24	7	4	5	9	6
25	7	4	7	1	2
26	8	6	1	7	6
27	8	6	5	1	4
28	8	6	7	3	2
29	8	6	11	7	4
30	9	8	1	9	2
31	9	8	5	3	4
32	9	8	11	9	6
1	0	0	1	1	2
2	0	0	7	7	6

–

The table above illustrates a continually repeating relationship between the whole numbers in each column on the spreadsheet, to which these “ending number columns”, that is the integer n, the times 12, and the ending number of the prime/non-prime column; correspond.

The first column has an ending digit that is now fixed in a rigid pattern; the second column of x12 has ending digits which are always double that in the first column: –

Even where the number 5 occurs in the first column, the ending digit shown in the second column is 0, which is actually representing a difference of 10.

It then follows that: 2 represents a difference of 12 in the second column (because in the 1^st. Col it is 6), 4 represents 14 (in the 1^st. Col it is 7), 6 represents 16 (in the 1^st. Col it is 8), 8 represents 18 (in the 1^st. Col it is 9).

The Conclusion on the Table of Matching Sequences in Relation to Gaps

The uplift sequence (1, 5, 7, 11) in column three is controlling the content results of the other columns and it is the gap sequence in column five that is controlling the complementary nature of the primes and non-primes.

The principal feature of these columns; the entire table and infinite part of the table that these series are for ever expandable to; – is that these patterns are immutable, ad infinitum.

Therefore I repeat; the fact that one Set of numbers (odd-composites as defined) – is exactly complimentary to another Set of numbers (prime numbers), I find more than fascinating considering that these two Sets of numbers can be originated completely separately.

Example from the Spreadsheet – (initial uplift of 5 is included)

There will be more detail on these relationships later.