PRIMES AND NON-PRIME PATTERNS -11 - Using Equation 2

Author – Andrew J Frost 10/06/2020 REV TWO PnonP p11

OBSERVATION 11 Using Equation 2

Equation 2 – by Sandor Kristyan –

My interpretation of above equation in to a spreadsheet formula is: – =IF(introw>=intcol,SUM(2*(2*((introw)*(intcol))+introw+intcol)+1),””) where intcol = a and introw = b. The ‘names’ – ‘introw’ and ‘intcol’ are denoting the infinite series of integers; which in this instance is the complete number line. -andy frost

What is interesting about this formula are the patterns produced – see the spreadsheet image below: –

The integers (number line) producing the spreadsheet results are in the 2^nd column and 4^th row.

Note:-

The blocks of 16 numbers (4 x 4) are surrounded by rows and columns of numbers ending in 5.
The blocks also include rows and columns of numbers that are divisible by 3.

There are also columns and rows with a background colour of blue – these are non-prime odd-composite numbers.

I’ll come back to these numbers again later.

The Spreadsheet of Eq2

Typical example blocks are shown with an orange boarder in the spreadsheet.

The result of n = 2(2ab+a+b)+1 where a>b
translated as: – =IF(introw>=intcol,SUM(2(2((introw)*(intcol))+introw+intcol)+1),””) where intcol = a and introw = b.

Ending Digits Block Pattern

The blocks contain a pattern of numbers such that the ending integers are, – (reading from left to right and down the columns): – 9,3,7,1 3,1,9,7 7,9,1,3 1,7,3,9 this is repeated without change.

This pattern is a mirror image (not the only mirror which will show up) of 2 rows and 2 columns, i.e. 9,3,7,1 3,1,9,7 | 7,9,1,3 1,7,3,9, where | is the mirror point. The diagonals in each set of 16 numbers end in the sequence 9,1,1,9 (top left to bottom right) or 1,9,9,1 (bottom left to top right) and repeat consistently.

However this pattern of ending numbers 9,3,7,1 3,1,9,7 | 7,9,1,3 1,7,3,9, is not in the same order as the ending digits pattern 17137939 first noted in Observation 4, which is the pattern that synchronises with the gaps pattern.

The Shape

On this spreadsheet, the cells with a green background show a special background shape, with the diagonal top edge sloping down the sheet (cells with green background and magenta borders) and the lower edge beneath curving back and down .

There is no limit to the extent of the non-prime odd-composite numbers that can be identified and added to the sheet calculation. However the Set of numbers, Set S_af retains its shape as an area on a “graph” (the cells with a green background) regardless of the extent of extension to the table by added integers.

Therefore what is the formula for this shape? What is the exact formula for identifying the numbers in Set S_af, which are the non-prime odd-composite numbers?

Div3 and Div5

Equation 2 generates rows and columns of numbers that are redundant for my purposes. I have identified the columns and rows that are redundant – these are the ones marked with “div3” and “div5” (numbers divisible by 3 and 5). When one analyses these, there is a pattern.

If the “div” lines and columns are hidden, then the remaining information is retained on rows and columns (the integers horizontally and vertically, named as ranges “introw” and “intcol”), that are spaced according to the following gap pattern: – 2,1,2,1,2,3,1,3, which then repeats. This of course is my pattern 4,2,4,2,4,6,2,6 divided by 2.

Columns associated with “div3 and div5” are not required.

Having hidden the redundant information, “div3” and “div5”; – the information that is left is just the prime and non-prime odd-composite number list, my list Set S_af. The rows and columns with a blue background are the remaining Set of non-prime odd-composite numbers. These numbers are composed of all the factors, which are themselves all combinations of prime numbers according to my previous definition.

Specific Integers

The “introw” and “intcol” sequences are actually ranges of sequences of eight integers of the values: – 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 24, 26, 29, 30, 33, …..

The first eight values are increasing by the above gap pattern – 2,1,2,1,2,3,1,3, to identify the following eight integer values (and so on) for both horizontal and vertical streams of integers. Each eight-number line of integers in the pattern is increasing by +15 from the first sequence shown, i.e. 3 + 15 = 18, 5 + 15 = 20 etc. based on the original sequence above.

These numbers I call my “Intcol Integer Sequence”.

2	1	2	1	2	3	1	3
3	5	6	8	9	11	14	15
18	20	21	23	24	26	29	30
33	35	36	38	39	41	44	45
48	50	51	53	54	56	59	60
63	65	66	68	69	71	74	75
78	80	81	83	84	86	89	90
93	95	96	98	99	101	104	105
108	110	111	113	114	116	119	120
123	125	126	128	129	131	134	135
138	140	141	143	144	146	149	150
153	155	156	158	159	161	164	165
168	170	171	173	174	176	179	180
183	185	186	188	189	191	194	195
198	200	201	203	204	206	209	210
213	215	216	218	219	221	224	225
228	230	231	233	234	236	239	240
243	245	246	248	249	251	254	255
258	260	261	263	264	266	269	270…etc

The Table of Integers Generated by the Gap Pattern

This pattern of integers then produces a spreadsheet table with only the non-primes odd-composite numbers that complement the prime numbers.

Duplication

For reasons that I do not understand at the moment, some of the non-prime odd-composite numbers are double and triple entries across the spreadsheet within this numbers landscape. (See end of post for spreadsheet link)

The formula derived from Equation 2 produces these duplicates and triplicate entries.

However what can be said about this duplication is that these numbers are odd-composites situated in the first column, with duplicates along rows (any rows). These rows terminate with a number that has a product of two or more primes (and only primes to my definition). The termination number of a row is either to the power of 2, 4 or 6… (or other greater even numbered exponent) or a terminating number that is the product of 2 or more sets of squares.

I suspect that one would find that every number in a row or in any column which this number is the head of, could be duplicated or a triplicate number. It would simply be a question of expanding this spreadsheet to enable the rows and columns needed to accommodate this. However since this would be an infinite expansion it is of course not feasible.

True Squares

Only columns headed by and rows ending in true squares or collections of true squares (diagonally left /right downwards) composed of prime factors excluding 2, 3, and 5, are required to complete Set S_af set of numbers.

It should be noted that initially the equation values of a and b, “intcol” and “introw” were started at 1.

It was at this point (today 20/08/2018) I tested for the inclusion of zero and negative numbers in this array.

(I had to insert the integer 2 into the rows, i.e. break my declared pattern; in order to get prime 5 to show in the list as a value for “a” in the formula of Equation 2.)

If zero is used in the equation; (intcol), instead of just producing non-prime odd-composite numbers, out pops a list of Prime numbers albeit still ‘polluted’ by the non-prime complementary numbers, which is exactly the list (i.e. Set S_af) created using my original method, my logical process formula.

This means that I am using Sandor Kristyan’s Equation 2 when the factors considered are only equal to or greater than 7.

The Spreadsheets

The spreadsheet is in my MediaFire account in the folder “Spreadsheet for PNonP Patterns” at the following address. Please copy and paste into your browser. The file sizes of the spreadsheets are – 32.55 Mb and 122.67 Mb.

You may download the spreadsheet file from this address: – https://www.mediafire.com/file/s5tof7b0edxpd38/20210101-PnonP-Main-orig20191130-Main.ods/file

Please let me know if you have a problem accessing this or any other spreadsheet.

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