Author – Andrew J Frost 10/06/2020 REV TWO PnonP p14. (Please note that while every effort is made to be clear about the information, it is better to read this series of posts from the beginning, i.e starting with the first post in the series “Primes and Non-Primer Patterns – 00” ).
OBSERVATION 14 – Gaps – The Random Nature of Gaps
Today (2019-01-25) Sheet: – https://www.mediafire.com/file/2alevqjpxu19bm0/20190209-p-non-p_set-af.ods/file
There are three sheets in the spreadsheet. (1.) P&nonP eq2 (2.) P&nonP all-int div3+5 eq2 (3.) PnonP S-af-int eq2 Squares of Primes Only
The integer sequence identified in Observation 13 that is producing the patterns under this previous heading are the result of my Observation 9. However if as described, as well as deleting the sieves of ‘div3’ and ‘div5’, I delete the rows that coincide with an odd-composite non-prime that occur in the result of My List (Saf) and Equation 2; then remove the corresponding columns with the same integer values; this will produce the integer sequence below. This now has a completely random gap sequence.
However whilst the gap sequence (gap) is randomised, the integer (int) sequence now corresponds to: – ref: OEIS A005097 (Odd primes – 1)/2.)
…int 0 3 5 6 8 9 11 14 15 | 18 20 21 23 26 29 30 33 | 35 36 39 41 44 48 50 51 | continued…
gap 3 2 1 2 1 2 3 1 | 3 2 1 2 132 3 1 3 | 2 1 231 2 3 143 2 1 |…
…int 53 54 56 63 65 68 69 74 | 75 78 81 83 86 89 90 95 | 96 98 99….
gap 2 1 2 3 173 2 132 1 1 3 231 2 132 3 1 352 | 1 2 1…
Red font above highlights – any number greater than or equal to three in the gap sequence and also is the resultant of two or more missing numbers, these being show as subscript either side of the red numbers. The positions of these red numbers above are indicated in the table below by the cells (non-primes) with a blue background.
See Table Below for comparison table taken from spreadsheet
Referring Back to the (int) Integer and the (gap) Gap Sequences
See spreadsheet link to my Mediafire Account at the top of this page. Looking at the third sheet “PnonP S-af-int eq2 Squares of Primes Only”, you will see that: –
> Following on from deleting the rows and columns above up to row and column 99, I have hidden further rows. These rows were identified by running a Conditional Format on the spreadsheet to identify duplicated numbers.
Running this format correctly highlights in light green background the duplicated numbers leaving the unique numbers. (Or correspondingly reversed by identifying non-duplicates with ‘grey’ conditional formatting as in Observation 12.) This is the “wave” or graph shape of the formula for planetary gravitation as previously noted.
These duplicated numbers are corresponding with odd-composite non-primes; having prime factors, which are primes from the same list; Set Saf. There is an organisation sequence to this series of prime factors which is noted already.
With further regard to the mirroring of the ending digits of these numbers which are all the squares of the primes and odd-composite non-primes in my list (n2 are 9,1,9,9, | 1,9,1,1,…); if then these numbers are themselves squared the series of ending digits is only 1. Integer 5 is excluded as previously.
There is a mirror point between the numbers in the “n” column at 73 | 77, which is consistent across the table through columns n2, n4 , and n6. This is in line with the ending digits of n2 and n4 between 5329 and 5929, 28398241 and 35153041, 151334226289 and 208422380089, respectively.
Using two digits the series is as follows: –
Table Showing n to a Power Exponent and the Two Ending Digits
n 1 (5) 07 11 13 17 19 23 29 31 37 41 43 47 49 53 59 61 67 71 73 77 79 83 89 91 97
n2 1 49 121 169 289 361 529 841 961 1369 1681 1849 2209 2401 2809 3481 3721 4489 5041 5329 5929 6241 6889 7921 8281 9409
n4 1 2401 14641 28561 83521 130321 279841 707281 923521 1874161 2825761 3418801 4879681 5764801 7890481 12117361 13845841 20151121 25411681 28398241 35153041 38950081 47458321 62742241 68574961 88529281
ending 2 digits n4 1 01 41 61 21 21 41 81 21 61 61 01 81 01 81 61 41 21 81 41 41 81 21 41 61 81
n6 1 117649 1771561 4826809 24137569 47045881 148035889 594823321 887503681 2565726409 4750104241 6321363049 10779215329 13841287201 22164361129 42180533641 51520374361 90458382169 128100283921 151334226289 208422380089 243087455521 326940373369 496981290961 567869252041 832972004929
The Ending Two Digits in n4
01, 41, 61, 21, 21, 41, 81, 21, 61, 61, 01, 81, 01, 81, 61, 41, 21, 81, 41, | 41, 81, 21, 41, 61, 81
The series of numbers is composed of – even numbered tens units plus one – as above (i.e. tens of 0, 2, 4, 6, 8). The odd numbered tens units (i.e. the first digit of the last two ending digits) do not appear in the list. If we then look at n to the power of 6 we see an ending digit sequence of numbers ending in 1 or 9 which has the same last single ending number pattern as n squared, but not the same last two ending numbers as n squared. However the mirror point occurs in the same position
There is no appearance of differences as far as I can detect that can be used as identification between numbers in the list.
SO FAR (at 2019-02-07)
There are various rigid patterns, but only when primes and their odd-composite complimentary numbers are considered together. Recapping on the information therefore: –
My Linux Calc interpretation of the above equation in to a spreadsheet formula is: – =IF(introw>=intcol,SUM(2*(2*((introw)*(intcol))+introw+intcol)+1),””).
n = 2(2ab+a+b)+1 where a>b, “Equation 2”
If intcol =0 then my set S(af) of numbers is produced. This includes all primes and all the special odd-composite complimentary numbers where the complimentary number is the product of two or more prime numbers from the primes in my set S(af). Each prime is used as a combination of two primes occurring only once in the Set of numbers, remembering that 2, 3, and 5 are not allowed as functioning primes in this series.
Therefore when b =0 (intcol =0) and a = the integer series (0), 3 5 6 8 9 11 14 15, 18… which is described by the (additions) gap sequence of 3 2 1 2 1 2 3 1 3 …. starting with 3, which is actually 0+3; then these series produce the correct sequence of integers by using the correct sequence of the gap series; my “Intcol Integer Sequence”. The sequence of integers is a pattern of 8 numbers continually increasing according to the gap series. Each Set of 8 numbers has an increase of 15 above the previous Set of numbers, thus
3 5 6 8 9 11 14 15,
18 20 21 23 24 26 29 30,
33………………ad infinitum – as shown in the table in Observation 11.
Then Equation 2, when b=0, reduces to 2a+1. This produces my set S(af); as long as the additions gap sequence is allowed to produce the integer sequence. Not identified by Sandor Kristyan, as far as I am able to comprehend his paper.
When a=b (intcol=introw), with the same gap and integer sequences in play for both a & b, where a prime is produced (ignoring odd-composites momentarily); then the resultant of Equation 2 is the square of the prime.
Where a & b do not produce a prime, but produce the special complimentary series of odd-composite numbers as in Set Saf, then the resultant is a number that is formed from a combination of squares and/or greater even powers of previous primes, acting as prime factors in my set (i.e. previously showing up in the order of the numbers).
For these odd-composite special complimentary numbers there is always a greater amount than 2 prime factors (i.e. greater than 1 and a prime}, in fact it will always be an even number of prime factors 4, 6, 8, and so on, except where there is only one unique prime factor, eg. 7 x 7 x 7 x 7 x 7 =75 = 16807, which is an odd-composite number produced by my set S(af).
See the spreadsheet.
This means that under these circumstances having developed the spreadsheet of –
n = 2(2ab+a+b)+1, where a = b, using the integer sequence (-Ref: OEIS A005097 (Odd primes – 1)/2); for both a and b; i.e. 3 5 6 8 9 11 14 15, 18…..(ad infinitum), the resultants are my special Set (S(af))2 . Therefore (S(af))2 = (2a + 1)2 , where a = any integer. Then (S(af))2 lets say equals ‘h’.
In 2(2ab+a+b)+1 if we substitute a for b since they are equal, we get 2(2a2 + 2a)+1, which is equal 4a2+4a +1 or 4a(a +1)+1, then when a=2, the resultant would be =25, for a=3 the resultant is =49, a=4 the resultant is =81, and so on. However a=2 and a=4 do not occur as they are not in the controlling sequence of identified integers.
A property of “h” is that the square-root of h produces a whole number every time, and the resultant factor(s) can be a prime; but only when the controlling integer sequence is 3 5 6 8 9 11 14 15, 18… When the resultant factor is not a prime; this indicates that it is an accumulation of further squares of odd-composite prime factor numbers from Set (S(af)) and these numbers are always a collection of factors that are prime factors in pairs or a single prime factor to the odd numbered power that have previously occurred in the list Set (S(af)). **
In other words the square-root of h is equal to (S(af)), when a = b, with a & b being the controlling sequence (OEIS A005097). Where a=0, in this particular column in the spreadsheet; the sequence (S(af)) is produced as a list and is equal to the equivalent square-root of (2a +1), which is a list of prime numbers in sequence and without gaps due to the controlling integer sequence. (See the spreadsheet).
Ref: OEIS A130290 Number of non-zero quadratic residues modulo the n-th prime. M. F. Hasler
Ref: OEIS A005097 (Odd primes -1)/2 N. J. A. Sloane (my note, | = start point of my sequence – three initial redundant series numbers are omitted [light grey]. This is the controlling integer sequence.)
1, 1, 2, | 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128…..
** There is another aspect of the prime factor numbers which I will come back to in a further post later.
Not So Cosmetic
Looking at these observations on this post and previous posts, I have agreed before that the information regarding patterns can be considered in some instances to be cosmetic, in that there are patterns, but that is all they are. However I would dispute that the patterns are so trivial and at every level. I do not believe this is just a cosmetic exercise.
Whilst I cannot express the patterns in anything but words, pictures and school mathematics; advanced maths and its concepts, symbolic methodology and representation, being something I find I cannot absorb nor use; I still believe I have come a long way.
There are post graduate papers out there that I have found since I started, and they are about the very concepts that I have expressed in these numbered posts (1 – 15 so far); but the maths in them is beyond me. However the concepts and content are the same; the conclusions, if any, are the same but arrived at in a way that only an advanced mathematician would understand what was written on the page.
I prefer to use English and pictures so that anyone can understand.
There are a number of observations expressed in my posts that have never been expressed before.
Spreadsheets
Spreadsheets are in my Mediafire account. (Sorry about the ads). Please click on the link at the top of this post or copy and paste this address into your browser.
