__(Also see Appendix A/Notes2016-2018/Section 4C/Pages 0000-50c, and thereafter: see Appendix A/Notes2016-2018/Sections 1 and 2/ ALL Pages for more. Appendix B will be the go-to appendix for BIM/24, PPTs and PRIMES, as well as the DSEQEC.)__

__(Also see Appendix Figures and Tables for BIM/24, PPTs and PRIMES, as well as the DSEQEC.)__

Rules 178-180 introduced the “sub-matrix grid” briefly as:

180 | BS Rule 180- The Sub-Matrix: Within the Inner Grid, every progressive number of every column or row is evenly divisible , progressively, by a whole number sequence. Plotting those dividends reveals a truly fundamental "sub-matrix" grid, underlying the original Inner Grid, of repeating simple whole number sequences, i.e. 1,2,3, ..., horizontally and vertically ...forming the simplest, most basic grid pattern possible, that also includes the axis numbers. All numbers of the original Brooks (Base) Square are predicated on this simple whole number sequence pattern. |

“Note: This amazing sub-matrix grid pattern explains the entire parent grid matrix and the **Inverse Square Law, ISL,** relationship that unfolds. See an animated example by CLICKING IMAGE above. Patience. Follow the bold numbers.

Here we find the simplest and most basic pattern of the simple whole number sequence ... 1,2,3,... forming a truly fundamental base layer ... a sub-matrix ... lying below the original grid. And every number on that original grid is predicated on this simple pattern of 1,2,3,... in both the horizontal and vertical arrays. In fact, we now have complete integration of the axis numbers with the Inner Grid numbers ... together forming the Prime Diagonal (PD) of the **Inverse Square Law** (**ISL**). We have come full circle. And it begs the question: Who is the parent and who is the child. It seems the parent has become the child. How many more fractal matrix layers are there?”

__Henceforth, we will call this Sub-Matrix 2.__

__A new sub-matrix, Sub-Matrix 1, results in the Active Row Sets (ARS) grid formed by dividing ALL Inner Grid numbers by 24.__

The simple — and some of the not so simple — geometries of the

**BIM**and**BIMtree**have been laid out in a series of interactive canvases. Each canvas opens up on its own webpage. Once open, you just click/touch and drag the geometry around, apart, this way and that way, even rebuilding the initial geometry side by side until you can really see how the parts are so simply related. It's deconstruction!The entire geometry on canvas series was produced initially without any supporting text to keep it as clean, simple and intuitive as possible. The accompanying text will certainly help fill in the gaps, but it should be noted that the geometry on canvas series sequentially builds from one canvas to another.

The algebraic binomials and their square proofs are widely known. Here the very same proofs overlap with that of the

**BIM**,**BIMtree**and the**ToPPTs**! This, too, can be readily seen.The number 24 — and its factors of 1-24, 2-12, 3-8, and 4-6 have shown, once again, to be integral to both the

**BIM**in general (remember, the EVEN Inner Grid cell numbers are all evenly divisible by 4), and, the**ToPPTs**. HOW SO? Well, this is new, really new, so just the surface has been touched, but so far:- Marking (YELLOW) all
**BIM**cells evenly ➗ by 24 generates a striking diamond-grid, criss-crossing pattern with 4 additional YELLOW marked cells in the center of each diamond; (*TPISC IV: Details*) - Every PPT is found to exist on —and ONLY on — those Rows whose 1st Column grid cell are ➗ by 24 (YELLOW, with 1st Column grid cell marked as '
**Z**'), though not every such YELLOW marked Row contains a PPT. nPPT are only present on such Rows if accompanied by a PPT; (*TPISC IV: Details*) - The “step-sister” of any given PT Row is found
**r**-steps down the Axis from that Row (with 1st Column grid cell marked as '**Z**') and it, too, always and only exists on a YELLOW marked Row (“_{f}**r**” is part of the FTP originally derived from the*Dickson Method*for algebraically calculating all PTs); (*TPISC IV: Details*) - The PPT Row (and nPPT Row) always contains the 4
**A**(**A**=area) value of that PT and the “step-sister” PPT Row always contains the 8**A**value, both landing exclusively on YELLOW marked grid cells, giving a striking visualization of ALL PPTs and their**r**-based “step-sisters;” (TPISC III: Clarity &*TPISC IV: Details*) - The significance of the “step-sister” is that it becomes the mathematical link to the “NEXT” PPT within the
**ToPPT**— like the Russian-Doll model; (TPISC III: Clarity) - The significance of the expanding and increasingly inter-connected PTs, as the
**BIM**itself expands, is one in which the perfect-symmetry geometry of regular shapes and solids — equal triangles, squares, circles,… of the**BIM**allows — at certain articulation nodes (i.e., Rows) — the introduction of the slightly less-perfect-symmetry geometry (i.e., bilateral symmetry) of the full rectangle and oval that the non-isosceles right triangle PTs represent, into the unfolding structural framework, working from the ground up, if you will. The roots of fractals-based self organization are first to grow here! (TPISC III: Clarity &*TPISC IV: Details*) more… - ALL self organization of any sort — be it force field or particulate matter — must have an organizing mathematical layer below driving it!

- Marking (YELLOW) all

Besides intimately tying the

**ToPPTs**to a natural fractal pattern within the**BIM**:- Thereafter, ALL PTs — primitive “parents” (PPT) and non-primitive “children” (nPPT) —are found on select Rows of the
**BIM**by simply following the squared numbers on the the Prime Diagonal down to intersecting Rows; ( TPISC I: Basics & TPISC II: Advanced) more… - The mathematical basis relating the PPTs was found, as was the consistent Fractal-Template Profile (FTP) that every PT follows; (TPISC II: Advanced & TPISC III: Clarity) more…
- A yet to be explained tie-in with the prime numbers and 24: The difference (∆) between the squares of ALL primes is evenly ➗ by 24!!! (Pattern in Number: from Primes to DNA; GoDNA: The Geometry of DNA; SCoDNA: Structure and Chemistry of DNA; Butterfly Primes: ~let the beauty seep in~; Butterfly Prime Directive: ~metamorphosis ~ ; and, Butterfly Prime Determinate Number Array (DNA): ~conspicuous abstinence ~; and an interactive Butterfly Primes new media net.art project.
- The FTP allowed all the PPTs to be sorted out and organized into a definitive
**Tree of Primitive Pythagorean Triples**(**ToPPT**) that co-extends infinitely throughout the infinitely expanding**BIM**—>**BIMtree**or**BIM-ToPPT**; (TPISC III: Clarity) more… and more…

One could simply say that: subtract one from the squared values of any natural, whole integer number (WIN) and if it is evenly ➗ by 24, it is a candidate for being a PPT or 'step-sister' if that same squared value - 25 is ALSO ➗ by 24.

If the Row contains BOTH a

**Factor Pair Set**(two squared values that =**a**and^{2}**b**) and the 4^{2}**A**value ONLY, it is a PPT Row.If it contains BOTH the 4

**A**and 8**A**values, it is a 'step-sister' Row, i.e. (**c**-1)/24 and (^{2}**c**-25)/24, if evenly divisible, are PPT and/or PPT 'step-sister' Rows, e.i. Row 17 is the PPT Row of the 8-15-17 PPT, but is also the 'step-sister' Row of the 5-12-13 PPT.^{2}(NOTE: The PPTs intersect with grid cells/24 at: the subtraction of 1,5,7,11,13,17,19,23,25,… from the squared Axis number (

**c**) and that gives a cell spacing of 4-2-4-2-4-2-4-2-… respectively, or a blank step ∆ between of 3,1,3,1,3,1,3,1,... )^{2}- Thereafter, ALL PTs — primitive “parents” (PPT) and non-primitive “children” (nPPT) —are found on select Rows of the
Take the

**BIM**and divide all numbers evenly divisible by 24.This gives you a criss-cross pattern based on 12, i.e. 12, 24, 36 ,48,… from Axis.

Halfway between, are the rows based on 6.

On either side of this 6-based and 12-based frequency, the rows just before and just after, are

**ACTIVE Rows**. These are ALWAYS ODD # Rows. They form an**Active Row Set**(**ARS**). Later colored in PURPLE.Their Axis #s are NEVER ➗3.They ALWAYS have their 1st Col value ➗ by 24.

Adding 24 to ANY of the ODD # NOT ➗3

**ACTIVE Row**Axis values ALWAYS sums to a value NOT ➗3 and thus to another**ACTIVE Row**Axis value (as adding 2 + 4 = 6, ➗3 added to a value NOT ➗3 = NOT ➗3 sum*).Another ODD Axis # Row lies before and after each pair of

**ACTIVE Rows**, i.e. between EVERY set of two**ACTIVE Rows**, is an ODD non-**ACTIVE Row**and their Col 1value is NOT ➗by 24.Adding 24 to ANY of these ODD # ➗3 Axis values ALWAYS sums to a value also ➗3 (as 2 + 4 = 6, ➗3 added to a value already ➗3 = ➗3 sum *).

While not an exclusive condition, it is a necessary condition, that ALL PPTs and ALL Primes have Col 1 evenly ➗ by 24.

Together, two

**ACTIVEs**+ one non-**ACTIVE**form a repetitive pattern down the Axis, i.e.**ARS**+ non-**Active Row**.*While 24 seems to define this relationship, any EVEN # ➗3 will pick out much if not all of this pattern, e.i., 6, 12, 18,…

It follows that:

ALL PTs (gray with small black dot) fall on an

**ACTIVE Row**.ALL PRIMES (RED with faint RED circle) fall on an

**ACTIVE Row**.The difference, ∆, in the SQUARED Axis #s on any two

**ACTIVE Rows**is ALWAYS divisible by 24.The difference, ∆, in the SQUARED Axis #s on an non-

**ACTIVE**ODD Row and an**ACTIVE Row**is NEVER divisible by 24.The difference, ∆, in the SQUARED Axis #s on any non-

**ACTIVE**ODD Row and another non-**ACTIVE**ODD Row is ALWAYS divisible by 24.Going sequentially down the Axis, every ODD number in the series follows this pattern:

nA—A-A—nA—A-A—nA—A-A—

#

__3__—5-7—__9__—11-13—__15__—17-19—__21__... Every 3rd ODD # (starting with 3) is ➗by 3 = nA .#

__5-7__—9—__11-13__—15—__17-19__Every 1st & 2nd, 4th & 5th, 7th & 8th,… ODD # is NOT ➗ by 3 = A.In other words, the two consecutive ODD #s, between the the nA ODD #s, are A ODD #s and are NOT ➗ by 3.

**#3,4,5 re-stated:**let A =**ACTIVE Row**Axis #, nA = non-**ACTIVE Row**Axis #A

_{2}^{2}-A_{1}^{2}= ➗ 24 and A ≠ ➗by 3nA

^{2}-A^{2}≠ ➗ 24nA

_{2}^{2}-nA_{1}^{2}= ➗ 24 and nA = ➗by 3##### 6,7,8 re-stated:

ODD Axis #s ➗by 3 (every 3rd ODD #) are NEVER

**ACTIVE Row**members — thus never PT/PRIMEODD Axis #s NOT ➗by 3 (every 1,2 — 4,5 — 7,8….ODD #s ) are ALWAYS

**ACTIVE Row**members and candidates for being PT and/or PRIME.

**In brief:** An ACTIVE Row ODD Axis # squared + a multiple of 24 (as 24x) = Another ACTIVE Row ODD Axis # squared , and the Square Root = a PT and/or a PRIME # :

A

_{1}^{2}+ 24x = A_{2}^{2}and √A_{2}^{2}= A_{2}= a PT and/or PRIME candidate; ODD

_{1}^{2}+ 24x = ODD_{2}^{2}and √ODD_{2}^{2}= ODD_{2}= a PT and/or PRIME candidate, if and only if, its 1st Col. value is ➗ by 24.__The difference in the squared values of any two PTs/PRIME #s (>3) is ALWAYS a multiple of 24!____On the Prime Diagonal, the ODD #s follow the same pattern as on the Axis (see No.7).__

See below under__BIM➗PPTs and PRIMES__: (Latest: as this work was being prepared, a NEW relationship was found.)*Why?*

Now that have overloaded the field with details, let's simply (again)!:

ALL PTs and ALL PRIMES are exclusively on Rows that:

• are referred to as “**ACTIVE**."

• bookend the Rows that are multiples (evenly ➗ by) 6: i.e. 6,12,18,24,30,36…

• some of these bookend on 6-based Rows do NOT have a PT and/or Prime, e.i. Rows 35, 49, 55, 77, 91, 95,… (Certainly, those ➗ by factors other than themselves and one, are NOT Prime.)

• as one progresses across the Matrix Row, the YELLOW and YELLOW-ORANGE cells on “**ACTIVE**” **Rows** follow a pattern under Column #: 1–5—7—11—13—17—19—23—25…i.e., + - - - + - + - - - + - + - - - + - + - - - and so on (see Gray-Violet & White on the Table below).

• the cell value numbers of the YELLOW and YELLOW-ORANGE cells on “**ACTIVE**” **Rows** follow exactly the numbers on **Col C** of the **Table VII**: Axis_Sqd_Diff_24x.numbers.

• the Table view gives the number values and relationships in tabular form, the Matrix in a more visual, geometric form.

• remember: ALL candidates for PTs and/or Primes MUST have **Col 1**/24 = (x2 - 1)/24 as TRUE.

Distilling the **BIM**/24 into the underlying "sub-matrix" of ➗24 **Actives**, as shown in **Figure below.** **__****_** , reveals exactly why the **BIM**/24 pattern is what it is.

NOTES: Sections 1-10

Section 1: 2018 (29pgs)

~~ ~~ ~~

In that the discovery that ALL PRIMES land only on **ARS** — exactly as do the PPTs — yet most emphatically do NOT follow the exact same distribution pattern leads to some open questions:

- what exactly is the relationship between PPTs and PRIMES?
- why do they BOTH land on
**ARS**? - why do some
**ARS**have both, neither, or one or the other? - can the PRIMES be used to predict the PPTs?
- can the PPTs be used to predict the PRIMES?

While a great deal in the way of proof of the PRIMES dependency on the **ARS** for their distribution will be presented in a number of tables **(Tables 1-23**, mostly redirected to the ** APPENDIX**), we must keep these questions in mind as we pursue the PPTs and their relationship to the

Now, we introduce

**Table 24**re-introduces the**Sub-Matrix 1**and how the PPTs (and PRIMES) are distributed strictly on the**ARS**.**Table 25**introduces the full**Sub-Matrix 2**, with the Columns 1,5,7,11,13,17,19,23,25,29,31,35,37,41,43,47, and 49 values shown in the colored inset boxes across the**BIM**.**Table 26**reduces**Table 25**down to a 5x1000**BIM**to focus primarily on the Col 1**Sub-Matrix 2**values (shown in the colored box insets).**Table 27**reduces this further to a 5x100**BIM.****Table 28**reduces this further to a 5x50**BIM.**

The details are the same:

These (colored inset boxes)

**Sub-Matrix 2**values:- ALL PPTs have Col 1 ➗4;
- NO PPTs have Col 1 NOT ➗4;
- For any given
**Active Rows Set**, only 1 Row is a ➗4 Row PPT, never both; - SOME Col 1 ➗4 Rows are NOT PPTs ( starred );
- The NOT PPTs ( starred ) Axis #s are ➗Prime Factors*.

Fool-proof Steps to Find ALL PPTs:

Axis# must be ODD, NOT ➗3 =

**Active Row Set**(**ARS**) member;Only 1 of the 2

**ARS**can be a PPT;**Sub-Matrix**Col 1 # MUST be ➗4;SOME may NOT be PPT if ➗Prime Factor (>5);

Remaining Axis # is a PPT. Exceptions:

Squared #s that are PPTs, remain PPTs when x2 or √x:

- ALL Squared #s that are PPTs, remain PPTs. ANY PPT #(x) times itself, times its square (x2) and/or times it serial products = NEW PPT;
- Example1: 5x5=25, 5x25=125, 5x125=625, 5x625=3125, 5x3125=15625=1252, 5x15625=78125, 5x78125=390625=6252,… products are ALL PPTs;
- Example 2: 97x97=9409, 97x9409=912,673, 97x912,673=88,529,281=94092=ALL PPTs.

Squared #s that are NOT, remain NOT when x2 or √x, as above.

**Table 29**Exponentials of the first 10 PPTs-values to be used in*c***Tables 30a-g**.**Tables 30a-g**The**Sub-Matrix 2**, when ➗4, and the difference (∆) between this and the next exponential PPT treated this way, is subsequently ➗ by its**Sub-Matrix 2**__variable__, the PREVIOUS exponential within the series is revealed. Restated as an example: When one subtracts 1 from the exponential values of(the*c*-value of the PPT) you get the*c***Sub-Matrix 2**value. Divide that by 4 and take the difference (∆) between it and the next. Divide that by 3 to give the PREVIOUS PPT-value in the series.*c*The

**Sub-Matrix 2**variable divisor = 3 =**Sub-Matrix 2**value/4 = 12/4. These variables run: 1,3,4,6,7,9,10,...

As to answers to the open questions called above:

what exactly is the relationship between PPTs and PRIMES?

- a loosely threaded connection is quite apparent;

why do they BOTH land on

**ARS**?- they both must be ODD #s, not ➗3, whose (x
^{2}-1)/24 is true;

- they both must be ODD #s, not ➗3, whose (x
why do some

**ARS**have both, neither, or one or the other?- some clouds, some clarity, at least for the PPTs;

can the PRIMES be used to predict the PPTs?

- yes, in the sense that if the PPT candidate is ➗Prime Factor (>5), it will not be a PPT;

can the PPTs be used to predict the PRIMES?

- currently, NO, yet the threaded connections are so great that the pattern will eventually emerge!

**SEE: Tables: 33a, 33b, 33c and 33d towards the end of Appendix B for some very NEW INFO on the BIM÷24.**

The underlying geometry of the

The

The interplay between these small sets of Numbers generates an incredible amount of richness and complexity with seemingly simplistic

Open in separate browser tab/window to see all.

A dovetailing of PPTs and PRIMES on the **BIM**

The discovery of the **Active Row Sets (ARS)** — the direct result of the ** BIM** ➗24 — in which it has been found that ALL PPTs and ALL PRIMES are exclusively found on, was in and of itself, a slow an arduous journey.

Once found, it has added a great deal of visual graphic clarity! In simple terms, it simply marks out the obvious. Both the PPTs and the PRIMES can not be on Axis Row #s that are EVEN, nor ➗3. This leaves ONLY Rows that are ODD #s and not ➗3.

The **BIM** ➗24 marked those **Active Row Sets** indirectly, by being on either side — i.e., +/- 1 — of the Axis Row # intercepted by the ➗24. Directly, the **ARS** was shown to be picked out by **Sub-matrix 1** and **2** values of the 1st cell Column of those Rows.

So we have the PPTs and the PRIMES occupying the same footprint rows, the **ARS** Rows. Both as a necessary, but not sufficient requirement, i.e., some **ARS** Rows do NOT have a PPT or PRIME, or both. ALL PPTs and ALL PRIMES are ALWAYS found on an **ARS** Row, NEVER on a **non**-**ARS** Row. Some **ARS** Rows may have none, either a PPT or a PRIME, or both.

Nevertheless, on this vast matrix array of **ISL** whole integer numbers, that the PPTs and PRIMES exclusively occupy the same ➗24-based footprint points to an underlying connection!

__The 1st connection__ was found and written about in the three white papers of 2005-6 on PRIMES:

- Page 2b:
**Butterfly Primes...let the beauty seep in.** - Page 2c:
**Butterfly Prime Directive...metamorphosis.** - Page 2d:
**Butterfly Prime Determinant Number Array (DNA) ~conspicuous abstinence~.**

__The 2nd connection__, as referenced below, has been the latest discovery that **Euler’s** **6n+1 and 6n-1 pick out, as a necessary — but not sufficient for primality — condition ALL the PRIMES.**

**When you look at the BIM**➗**24, you can readily see how this theorem simply picks out the very same ARS Rows! (For ARS 5 and above.)**

**The BIM**➗**24 becomes a DIRECT GRAPHIC VISUALIZATION of EULER’s PRIMES = 6n+1 and 6n-1, where n=1,2,3,..**

The same holds true for **Fermat's (Fermat-Euler)** **4n + 1** = **Sum of Two Squares Theorem**, where 4n + 3 ≠ Sum of Two Squares. The **4n + 1** = **Sum of Two Squares** = **Pythagorean Primes** (PTs where ** c** = Prime #).

These are simply those **AR**s that contain BOTH a PRIME (red circle) AND a PPT (black dot) in the figures.

Note: In any given **ARS**, only one of the two **AR**s may be a PPT, while both, neither, or one or the other **AR**s may be PRIME.

Black dot in a Red circle = **Pythagorean Triple** = **4n + 1 PRIME candidate** = **Sum of Two Squares**.

__The 3rd connection__ is that **for those ARS Rows that do NOT contain PRIMES — e.i., 25, 35, 49, … and have been shown to negate the possibility of the # being prime because it is itself prime factorable — divisible by another set of primes — is ALSO DIRECTLY VISUALIZABLE ON THE BIM**➗**24 AS THE INTERSECTING PRIME COLUMNS!!!**

__The 4th connection__ reveals that the **BIM Prime Diagonal (PD) — the simple squares of the Axis #s — defines:**

**The ISL itself, as every BIM Inner Grid cell value is simply the difference between its horizontal and vertical PD values;****The Pythagorean Triples, as every PD cell value points to a PT when one drops down from it to its ARS intercept;****The PRIMES, as the difference in the squares of any two PRIMES (≧5) — every PD value of a PRIME Axis # — is evenly ➗24.**

__The 5th connection__ is that **for those ARS Rows— that may or may not contain PPTs and/or PRIMES — their 1,5,7,11,13,17,19,23,25,.. ODD intersecting Columns NOT ➗3, ARE ALL➗24, and, this is ALSO DIRECTLY VISUALIZABLE ON THE BIM**➗**24 AS THE INTERSECTING COLUMNS (usually depicted graphically in YELLOW-ORANGE boxes/cells on the BIM as part of the diamond with centers pattern) !!!**

￼ This is revealed in the **Sub-Matrix 1** figures and tables: the **BIM** ÷24.

Additionally, **Sub-Matrix 2** also selects for ALL **ARS** as Column 1 ALWAYS ÷4. (See figures.)

In ALL cases, the PRIMES (≧5) are necessary — but not sufficient to insure primality — located on ODD # Rows NOT ÷3. It is as simple as that!

The factors of 24 — 1,24–2,12–3,8–4,6 — when increased or decreased by 1, ultimately pick out ALL **AR**s. **Euler's 6n +/-1** is the most direct, **Fermat's 4n + 1** gets the **Sums of Two Squares** = **Pythagorean Primes** (while 4n + 3 gets the rest).

**Fermat's Little Theorem** (as opposed to the more familiar "Fermat's Last Theorem") tests for primality.

But now there is a dead simple way to test for primality:

**The difference in the squares between ANY 2 PRIMES (≧5) ALWAYS = n24.**

For example, take any random ODD # - 25 —> it must be ÷24 n times to be PRIME. (n=1,2,3,...)

741

741

^{2}-5^{2}/24 = n= 22877.3 NOT PRIME189

189

^{2}-5^{2}/24 = n = 1487.3 NOT PRIME**289**289

^{2}-5^{2}/24 = n = 3479**PRIME**

￼

**TPISC, The Pythagorean-Inverse Square Connection, has evolved into a connection with the PRIMES (TPISC-P).**

- Goldbach (Strong) Conjecture (every even # is made of two primes)
- Goldbach Tertiary (Weak) Conjecture (every odd # is made of three primes)
- Twin Prime Conjecture (there are an infinite # of primes separated by 2)
- Twin Primes Conjecture (there are an infinite # of primes separated by a fixed # gap)
- Prime Gap (size between consecutive primes)
- Prime Triple Conjecture (there are an infinite # of 3 consecutive primes with ∆ of 6, first and last)
- Prime Quadruple Conjecture (there are an infinite # of 4 consecutive primes with ∆ of 8, first and last)
- Prime k-tuplet Conjecture (there are an infinite # of prime
*k*-tuplets for each* k*) - Dickson’s Conjecture (there are a lot of primes, Twin, Sophie Germain,
*k*-tuplet,…) - Pythagorean Primes Conjecture(Pythagorean triples with prime # hypotenuse)

*(1 above.)*This was examined and a proof-solution was offered in 2010 with the Periodic Table of Primes (PTOP), hidden within the**BIM**.(

*2 above.)*Work remains.*(3 above.)*ALL Primes fall on**Active Rows (ARs)**within an**Active Row Set (ARS)**of three Axis #s: two ODDs with an EVEN in-between. The difference (∆) between the ODDs = 2. Thus ANY and ALL Twin Primes — separated by 2 — are seen right here, and only here, directly, on the**ARS**.*(4-8 above.)*Because the**ARSs**follow a strict**Number Pattern Sequence (NPS)**in that the EVEN #s are ÷12 — being the endpoint of**BIM÷24**— and the bookcased ODD #s are NEVER ÷3, there is a built-in ∆, a natural gap, between the**ARS and ODD #s ÷ 3.**This means the**ARs**— that provide a necessary, but not sufficient requirement for ANY PRIME — must necessarily have gaps that are:__taken from the lower__**ARS**#:- (Twins of) 2, 6, 8, 12, 14, 18, 20,… i.e., up 2, 4, 2, 4, 2, 4, 2,….

__taken from the higher__**ARS**#:- (Twins of) 4, 6, 10, 12, 16, 18, 22,… i.e., up 4, 2, 4, 2, 4, 2, 4,….

*(9 above.)*Dickson’s Conjecture leads to many of the individual conjectures above. One, the Sophie Germain Prime Conjecture states that if a PRIME #,*p*, has another PRIME # generated at 2*p*+ 1, it is a Sophie Germain PRIME and that there are an infinite number of these. On the**BIM**, it is easy to see that any Sophie PRIME is simply taking the lower**AR**# and ADDING a multiple of 6 — 6x — to get to the next PRIME. Try it: (≧5)5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, ... OEIS: A005384. (https://en.wikipedia.org/wiki/Sophie_Germain_prime)*the lower

**AR**+ 2 = higher**AR**— neither is ÷ by 3, while the lower**AR**- 2 = ODD# that is ÷ by 3 and NOT an**AR**.This works for

*p*≧5:5+6=11, 11+(2)6=23, 23+6=29, 29+(2)6=41, 53+(5)6=83, 83+6=89, 89+(4)6=113, 113+(3)6=131, 131+(7)6=173, 173+(1)6=179, 179+(2)6=191, 191+(7)6=233, 233+(1)6=239, 239+(2)6=251, 251+(5)6=281, 281+(2)6=293, 293+(11)6=359, 359+(10)6=419, 419+(2)6=431, 431+(2)6=443, 443+(8)6=491, 491+(3)6=509, 509+(14)6=593, 593+(8)6=641, 641+(2)6=653, 653+(1)6=659, 659+(4)6=683, 683+(6)6=719, 719+(4)6=743, 743+(3)6=761, 761+(8)6=809, 809+(17)6=811, 911+(7)6=953, …

It appears that this conjecture really is simply restating Euler’s 6

*n*+1 and 6*n*-1 — where*n*=1,2,3,… method for determining ALL PRIMES.*(10 above.)*Ironically, the Pythagorean Triples were first to be found on the**ARS**, and only after reviewing earlier work — the Butterfly Primes — that the connection of the PRIMES to the**BIM÷24**was made. Both the Primitive Pythagorean Triples (PPTs) and the PRIMES have the same — again, necessary, but not sufficient requirement — that they occupy EXCLUSIVELY the**ARs**potentially of any**ARS**.__When the PPTs overlap the PRIMES, we get the Pythagorean Primes.__An interesting note is that unlike the PRIMES that can occupy BOTH**ARs**of a given**ARS**, the PPTs will NEVER do this, occupying either one or the other, but NEVER BOTH,**ARs**within a**ARS**.Fermat's (Fermat-Euler) 4

*n*+ 1 = Sum of Two Squares Theorem, where 4*n*+ 3 ≠ Sum of Two Squares. The 4*n*+ 1 = Sum of Two Squares = Pythagorean Primes (PTs where the hypotenuse,*c*= Prime #).Remember, not only does the

**BIM÷24**reveal the**ARs**within the**ARS**, the**∆ BETWEEN THE SQUARES OF ANY PAIR OF AR Axis #s (**≧**5) IS A MULTIPLE OF 24**. The latter becomes a necessary, but not sufficient test of both primality, and, PPT validity.Note that this same ∆ occurs with pairs of two

**non-AR**ODD #s, i.e. 9 and 15. Both are ÷3. 152 - 92 = 144, and 144/24 = 6. A hybrid of an**AR**and a**non-AR**ODD pair set will NOT show a ∆ of 24x.This points to underlying

**NPS**within the**BIM**(see the pattern below the links):__For the ODD #s__^{2}:∆÷24 vs NOT ∆÷24

∆÷24 in

**ARS**vs ∆÷24 in**non-ARS**, NEVER a mix of the two__For the EVEN #s__:^{2}∆÷24 vs NOT ∆÷24

∆÷24 includes those in progressive

**NPS**series ∆2, ∆4, ∆6, ∆8, ∆10,… (see Table in Appendix ).The PPTs and PRIMES are STRICTLY following this

**NPS**within the**BIM**!Perhaps the biggest finding here is:

! Just like the PPTs!__The PRIMES — Inverse Square Law Connection__**TPISC**stands for both, AND,*The Pythagorean-Inverse Square Connection*. TP-P-ISC, TP/P-ISC, TP/PISC, TPPISC, or TP-IS-PC,...*The PRIMES-Inverse Square Connection***The connection between that Universal Law, the****ISL**, — the underlying law of ALL of spacetime —and the PRIMES — the fundamental "quarks" of the number quantity system, AND, the Pythagorean Triples — the fundamental right-triangle/rectangle form of that same geometry — is without out a doubt the most intriguing, beguiling, and misunderstood relationship we are only just NOW getting a real glimpse at. The future looks very promising!

**"A New Kind of Prime**

The twin primes conjecture’s most famous prediction is that there are infinitely many prime pairs with a difference of 2. But the statement is more general than that. It predicts that there are infinitely many pairs of primes with a difference of 4 (such as 3 and 7) or 14 (293 and 307), or with any even gap that you might want."

Quote is from Quanta Mag 9/26/19 article: Big Question About Primes Proved in Small Number Systems

by Kevin Hartnett

- Twin Primes
- Primes separated by other EVEN numbers
- Euler's 6
*n*+1 and 6*n*-1 - Fermat-Euler's 4
*n*+ 1= Sum of Two Squares Theorem (Pythagorean Primes) - Dickson's Conjectures: Sophie Germain Primes
- Goldbach Conjecture (Euler's "strong" form)
- Primes - BIM - Pythagorean Triples

~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~

A new and easy visualization!

There is a new game board in town. It has actually been around for awhile, but few know of it. It's basically a matrix grid of natural numbers that defines the Inverse Square Law (ISL). It's called the BIM, short for the BBS-ISL Matrix. Every grid cell is uniquely occupied by a given number that is simply the difference between the horizontal and vertical intercept values of the main "Prime Diagonal" (not Prime number diagonal) that mirror-divides the whole matrix.

If you show all the matrix values that are evenly divisible by 24, a criss-crossing pattern of diamond-diagonal lines will appear and this ends up giving us a unique visual on the distribuition of ALL PRIMES!

Basically, all ODDs (3 and greater for this presentation) fall into a repetitive pattern of:

ODD ÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3...

The matrix ÷24 described above, nicely picks out these sets of —Not÷3—Not÷3 that we call Active Row Sets (ARS). Each ODD member of the ARS is referred to as an Active Row (AR).

The ÷3 ODD Rows that lie between are called Non-Active Rows, or NA.

It is a necessary, but not sufficient condition, that ALL PRIMES are strictly located on ARs. NO exceptions, except for some ARs have NO PRIMES.

Another striking visualization is that ALL Primitive Pythagorean Triples (PPTs) also lie only on the ARs, following the same necessary, but not sufficient, condition. NO exceptions here as well, except for some ARs also have NO PPTs. In addition, only one of the ARs within an ARS can have a PPT.

So any AR within an ARS can have 0/1 PRIMES and/or 0/1 PPTs in any combination, only with the one caveat: that only 1 PPT/ARS is allowed.

Now, not to belabor this BIM game board, let's talk PRIME GAPS!

Let's refer to the lower number value for the ARS as "Lower" and the higher as "Upper." All are ODDs. As will become apparent, you can always tell if your ODD AR is "Lower/Upper" simply by adding 2 and ÷3: if the result evenly ÷3 it is "Upper" and if not, "Lower."

The Even Gap between primes strictly follows this pattern: (all are necessary but not sufficient conditions for primality)

For any given set of twin primes, or any single prime, that is the smaller "Lower" of an AR set:

- Even Gap must be 2, 6, 8, 12, 14, 18, 20, 24, 26, 30, 32, 36, 38, 42, 44, 48, 50,…
- Example: P=5: 7, 11, 13, 17, 19, 23, (25), 29, 31, (35), 37, 41, 43, 47, (49), 53, (55)
- P + 2, P + 2+4, P + 2+6, P + 2+10, P + 2+12, P + 2+16, P + 2+18, P + 2+22, P + 2+24,…
- P+2
^{1}, P+2+2^{2}, P+2^{3}, P+2^{2}+2^{3}, P+2+2^{2}+2^{3}, P+2+2^{4}, P+2^{2}+2^{4}, P+2^{3}+2^{4}, P+2+2^{3}2^{4} - The Even Gap difference pattern follows: 2-4-2-4-2-4-….

For any given set of twin primes, or any single prime, that is the larger "Upper" of an AR set:

- Even Gap must be 4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 34, 36, 40, 42, 46, 48, 52, 54.…
- Example: P=7: 11, 13, 17, 19, 23, (25), 29, 31, (35), 37, 41, 43, 47, (49), 53, (55), 59, 61
- P + 4, P + 2+4, P + 2+8, P + 2+10, P + 2+14, P + 2+16, P + 2+20, P + 2+22, P + 2+26, P + 2+28,
- P+2
^{2}, P+2+2^{2}, P+2+2^{3}, P+2^{2}+2^{3}, P+2^{4}, P+2+2^{4}, P+2+2^{2}+2^{4}, P+2^{3}+2^{4}, P+2^{2}+2^{3}+2^{4}, P+2+2^{2}+2^{3}+2^{4}, - The Even Gap difference pattern follows: 4-2-4-2-4-2-…. and is identical in the middle, not the start!

One can readily see that all this is simply the natural result of two ARs alternating with a NA row:

AR—AR—NA—AR—AR—NA—AR—AR—NA—AR—AR—NA—…

**“Lower” AR number**

**AR—5**

AR—7

NA—9

**AR—11**

AR—13

NA—15

**AR—17**

AR—19

NA—21

**AR—23**

AR—25

NA—27…

It is 6 steps, i.e. +6, from any NA to the next NA, from any “Lower” AR to the next “Lower” AR, from any “Upper” AR to the next “Upper” AR. ALL of the various EVEN GAPS may be shown to be a direct consequence of the natural number sequence and easily visualized on the BIM.

To any “Lower” Active Axis Row ODD:

- Add 2, 4, or 6,…
- If sum is ÷3, it is a NA
- If sum is NOT÷3, it is an AR number and a PRIME (or PPT) candidate.

**“Upper” AR number**

**AR—7**

NA—9

AR—11

**AR—13**

NA—15

AR—17

**AR—19**

NA—21

AR—23

**AR—25**

NA—27…

To any “Upper” Active Axis Row ODD:

- Add 2, 4, or 6,…
- If sum is ÷3, it is a NA
- If sum is NOT÷3, it is an AR number and a PRIME (or PPT) candidate.

Again, It is 6 steps, i.e. +6, from any NA to the next NA, from any “Lower” AR to the next “Lower” AR, from any “Upper” AR to the next “Upper” AR. ALL of the various EVEN GAPS may be shown to be a direct consequence of the natural number sequence and easily visualized on the BIM.

The ARS pattern on the BIM clearly shows the above patterns and may be extrapolated to infinity:

A number of PRIME conjectures have been shown to be easily visualized on the BIM:

Reference: TPISC IV: Details: BIM + PTs + PRIMES

The factors of 24 — 1,24–2,12–3,8–4,6 — when increased or decreased by 1, ultimately pick out ALL **AR**s. **Euler's 6 n +/-1** is the most direct,

**Fermat's Little Theorem** (as opposed to the more familiar "Fermat's Last Theorem") tests for primality.

But now there is a dead simple way to test for primality:

**The difference in the squares between ANY 2 PRIMES (≧5) ALWAYS = n24.**

For example, take any random ODD # - 25 —> it must be ÷24 *n* times to be PRIME. (*n*=1,2,3,...)

741

7412-52/24 =

*n*= 22877.3 NOT PRIME189

1892-52/24 =

*n*= 1487.3 NOT PRIME**289**2892-52/24 =

*n*= 3479**PRIME**

I would be remiss if I did not mention that throughout this long journey that began with the Inverse Square Law and the Primtive Pythagorean Triples that the PRIMES kinda of just fell out. They just kept popping up. Not the least, but certainly not without effort, the BIM actually directly visualizes ALL of the NO-PRIMES. In doing so, one is left with information that is simply the inverse of the PRIMES. Subtract the NO-PRIMES information from the list of ODDS (disregarding 2) and what remains are ALL the PRIMES! A simple algebraic expression falls out from that:

NP = 6*yx* +/- *y*

letting *x*=1,2,3,… and *y*=ODDs 3,5,7,… with the only caveat is that if you don't first eliminate all the ÷3 ODDS, you must include exponentials of 3 (3^{x}) in the NP tally.

In 2009-10, a solution to Euler's "strong" form of the Goldbach Conjecture "that every even positive integer greater than or equal to 4 can be written as a sum of two primes" was presented as the BBS-ISL Matrix Rule 169 and 170. This work generated a Periodic Table of Primes (PTOP) in which Prime Pair Sets (PPsets) that sequentially formed the EVEN numbers were laid out. It is highly patterned table. A recently annotated version is included with the original below. (See above link for details.)

It turned out this PTOP was actually embedded — albeit hidden — within the BIM itself as shown in Rule 170 and here, too, a recently annotated version is included below. (See above link for details.)

**Rule 169**: Periodic Table of Primes.

**Rule 169**: annotated

**Rule 170**: Periodic Table of Primes (PTOP): embedded within Brooks (Base) Square.

**Rule 170**: annotated

MathspeedST: TPISC Media Center

BACK: ---> Simple Path BIM to PRIMES on a separate White Paper BACK: ---> PRIMES vs NO-PRIMES on a separate White Paper BACK: ---> TPISC_IV: Details_BIM+PTs+PRIMES on a separate White Paper BACK: ---> PRIME GAPS on a separate White Paper (This work.)

A NEW UPDATE: scroll down below these references!

**Additional useful links:**

See **Appendix B: Tables 31-33** for more info.

https://en.wikipedia.org/wiki/Goldbach%27s_conjecture

https://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture

https://en.wikipedia.org/wiki/Pythagorean_triple

https://en.wikipedia.org/wiki/Twin_prime

https://en.wikipedia.org/wiki/Prime_gap

https://en.wikipedia.org/wiki/Prime_triplet

https://en.wikipedia.org/wiki/Prime_quadruplet

https://en.wikipedia.org/wiki/Prime_quadruplet#Prime_quintuplets

https://en.wikipedia.org/wiki/Prime_k-tuple

https://en.wikipedia.org/wiki/Dickson%27s_conjecture

https://en.wikipedia.org/wiki/Sophie_Germain_prime

https://en.wikipedia.org/w/index.php?search=Pythagorean+Primes&title=Special%3ASearch&go=Go

https://en.wikipedia.org/wiki/Prime_k-tuple

https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares

https://en.wikipedia.org/wiki/Pythagorean_prime

https://en.wikipedia.org/wiki/Fermat%27s_little_theorem

https://en.wikipedia.org/wiki/Euler%27s_theorem

https://en.wikipedia.org/wiki/Euclid–Euler_theorem

http://mathworld.wolfram.com/Eulers6nPlus1Theorem.html

Earlier we found:

__For the ODD #s ^{2}__

- ∆÷24 vs NOT ∆÷24
- ∆÷24 in
**ARS**vs ∆÷24 in**non-ARS**, NEVER a mix of the two

__For the EVEN #s ^{2}__

- ∆÷24 vs NOT ∆÷24
- ∆÷24 includes those in progressive
**NPS**series ∆2, ∆4, ∆6, ∆8, ∆10,… (see Table in Appendix ).

The PPTs and PRIMES are STRICTLY following this **NPS** within the **BIM**!

**Let’s dispense with the EVENs first.**

If one takes the EVEN #s on either side of the

**ARS**+ the EVEN # in the MIDDLE of the**ARS**, we have a new set. Let’s call it the**EVENs Set (ES)**.Well, it turns out there are two versions of the

**ESs**and they alternate down the PD. Let’s call them**ES**and_{1}**ES**and they go:_{2}**ES**-_{1}**ES**-_{2}**ES**-_{1}**ES**,…_{2}If we plot the

**ESs**along the PD it will look like this (remember these are the squares of the EVEN Axis #s):16 —

**36**— 64 100 —**144**— 196256 —

**324**— 400 484 —**576**— 676784 —

**900**—1024 1156—**1296**—14441600—

**1764**—1936 2116—**2304**—2500The

**ES**are on the Left, the_{1}**ES**are on the Right._{2}

The ÷24 **NPS** of the **ES _{1}** and

Both group sets bookend the

**ARS**with an ODD #÷3 (**NON-ARS**ODD) between.The OUTER ends of

**ES**are ∆/24 and the OUTER ends of_{1}**ES**are ∆/24._{2}The MIDDLE of

**ES**are ∆/24 and the MIDDLE of_{1}**ES**are ∆/24._{2}There is NO MIXING between sets and NO MIXING OUTER with MIDDLE #s.

- The MIDDLE
**ES**is ÷ by the 1st MIDDLE of the whole_{1}**ES**set._{1} - The MIDDLE
**ES**is ÷ by the 1st MIDDLE of the whole_{2}**ES**set._{2}

- The MIDDLE
**ES**÷ 16 ÷ 8 ÷ 4 ÷ 2 OUTER; MIDDLE ÷ 36, NEVER ÷ 24. (NOTE: these are the PD#s and not the ∆ in PD#s.)_{1}**ES**————÷ 4 ÷ 2 OUTER; MIDDLE ÷ 36, ALWAYS ÷ 24. (NOTE: these are the PD#s and not the ∆ in PD#s.)_{2}A look at the

**BIM ÷24**shows that ALL MIDDLE PDs of their**ESs**are ÷12 and graphically:Table 31d. Snapshot of the PDF showing the EVENS role in the**BIM ÷24**Distribution.**ES**fall along the IN-BETWEEN YELLOW DIAMOND PATTERN formed from the_{1}**BIM ÷24**.- NEVER ÷ 24.

**ES**fall at the Axial POINTS of the YELLOW DIAMOND PATTERN._{2}- ALWAYS ÷ 24.

- ALWAYS ÷ 24.

Table31d PDF sequence. The EVENS. Open the PDFs in a separate tab/window to see all pages.

Table 33c. The EVENS

So now that we have established that the EVENS, too, reveal a ÷24 expression within the **BIM**, let's move on the ODDS.

__What about the ODDs?__

We know that the ODDs ÷3 are the separators of the EVEN's **ES _{1}** and

We will continue to label the two groups as **ARs** and **NON-ARs**.

Earlier we found:

For the ODD #s^{2}

- ∆÷24 vs NOT ∆÷24
- ∆÷24 in
**ARS**vs ∆÷24 in**non-ARS**, NEVER a mix of the two.

More precisely:

ALL

**ARs**^{2}are ∆/24, and, NEVER ÷3, and after subtracting (-1) are ALWAYS ÷24, and, ALL now ÷3.ALL

**NON-ARs**are ∆/24, and, ALWAYS ÷3, and after subtracting (-1) NEVER ÷24, but ALL ÷9 ÷3.^{2}There is NO MIXING between groups.

Additionally, the ODD

**AR**group may be further divided into:**ODD Set 1 (OS**: (Like the EVENS,_{1})**ES**) for IN-BETWEEN YELLOW DIAMOND Rows 6,18,30,42,..._{1}**ODD Set 2 (OS**: (Like the EVENS,_{2})**ES**) for the POINTS of the YELLOW DIAMOND Rows 12,24,36,..._{2}However, here, BOTH groups ARE MIXABLE, i.e. ∆/24 across

**OS**and_{1}**OS**._{2}What really becomes interesting about the ODDs on the

**BIM ÷24**is that, all by themselves without regard to the PPTs or PRIMES, they form a deep, underlying**NPS ALL BASED ON THEIR ∆s/24**!In a nutshell, if you take each ODD #, square it (that's the PD#) and now sequentially substract the ODD PDs below it, and then ÷24, the results will generate a tightly connected

**NPS**across the ODDS on the**BIM**.The

**ARs**1-25-49-121-169-289-361-529-625,… (1^{2}^{2}-5^{2}-7^{2}-11^{2}-13^{2}-17^{2}-19^{2}-23^{2}-25^{2},) subtraction=**ALL ARs**.The

**NON-ARs**9-81-225-441-729,… (3^{2}^{2}-9^{2}-15^{2}-21^{2}-27^{2},…) subtraction=**ALL NON-ARs**.

Here is a quick look at the AR and NON-AR Distribution. Open the PDFs below in a separate tab/window.

Here is a quick look at the AR and NON-AR Distribution. Open the PDFs below in a separate tab/window.

Table 33a. ODDs AR and NON-AR Distribution PDF. Open the PDFs in a separate tab/window to see all pages.

Table 33b. ODDs sequenced PDF. Open the PDFs in a separate tab/window to see all pages.

Table 33c ODDs_more detail PDF. Open the PDFs in a separate tab/window to see all pages.

Table 33d-2-1

Table 33d-2-2

Table 33d-2

Back to the Table 31 sequences. Table 31a... series, along with the previous Table33 series, uncovers how the PRIMES were found within the **BIM** both by algebra and geometry. This has been reduced to a focus white paper: "** PRIMES vs NO-PRIMES** that is a condensed summary of this work. One may benefit from looking over this paper prior to the study of these more elaborate tables. A summary from the paper is presented here:

The BIM is a symmetrical grid — divided equally down its diagonal center with the Prime Diagonal (PD) — that illuminates the Number Pattern Sequence (NPS) of the Inverse Square Law (ISL) via simple, natural Whole Integer Numbers (WIN).

The BIM Axis numbers are 1,2,3,.. with 0 at the origin.

The Inner Grid (IG) contains EVEN and ODD WIN, but except for the 1st diagonal next to the PD — a diagonal that contains ALL the ODD WIN — there are NO PRIMES (NO-PRIMES, NP) on the SIG (Strict Inner Grid).

The PD WIN are simple the square of the Axis WIN.

ALL the IG WIN result from subtracting the horizontal from the vertical intersection of the PD.

Dropping down a given PD Squared WIN (>4) until it intersects with another squared WIN on a Row below will ALWAYS reveal that Row to be a Primitive Pythagorean Triple (PPT) Row, whose hypotenuse,

*c*, lies on the intersecting PD. ALL PPTs may be identified this way.Dividing the BIM cell values by 24 — BIM÷24 — forms a criss-crossing DIAMOND NPS that divides the overall BIM into two distinct and alternating Row (and Column) bands or sets:

- ODD WIN that are ÷3 and referred to as NON-ARs;
- ODD WIN that are NOT ÷3 and referred to as ARs, or Active Rows;
- The ARs ALWAYS come in pairs — with an EVEN WIN between — as the UPPER and LOWER AR of the ARS (Active Row Set);

ALL PPTs and ALL PRIMES ALWAYS are found exclusively on the ARs — no exceptions.

By applying:

(1) *

let

*y*= odd number (ODD) 3, 5, 7,… and*x*= 1, 2, 3,... one generates a NP table containing ALL the NP;*True if ÷3 ODDs are first eliminated, otherwise ADD exponentials of 3 to the NP pool;

Eliminating the NP — and the NP contain a NPS — from ALL the ODD WIN, reveals the PRIMES (P).

A necessary, but not sufficient confirmation — but not proof — of primality is found by finding the even division of 24 into the difference of the square of ANY two PRIMES as:

(2) (P

_{2})^{2}- (P_{1})^{2}=let

Be aware that this also holds true for ALL the AR NP. The P and NP are NOT ÷3, and are both part of the ARS and therefore any combination of the two squared differences will be ÷24:

(3) (NP

_{2})^{2}- (NP_{1})^{2}=(4) (NP

_{2})^{2}- (P_{1})^{2}=(5) (P

_{2})^{2}- (NP_{1})^{2}=The ÷3 NON-AR set is separately ÷24, but can NOT be mixed with members of the AR set (ARS) as:

(6) (NON-AR-NP

_{2})^{2}- (NON-AR-NP_{1})^{2}= n24(7) (NON-AR-NP

_{2})^{2}- (NP_{1})^{2}⧣(8) (NP

_{2})^{2}- (NON-AR-NP_{1})^{2}⧣(9) (P

_{2})^{2}- (NON-AR-NP_{1})^{2}⧣(10) (NON-AR-NP

_{2})^{2}- (P_{1})^{2}⧣The division into AR and NON-AR sets has a NPS that ultimately define the elusive pattern of the P.

Furthermore, may be re-arranged to:

(11)

(12)

asking whether any given ODD (>3) is a P or NP, it is exclusively a NP if, and only if,

*y*reduces to the same value after applying*x*. As*y*is effectively an ODD of either a PRIME or composite of PRIMES factor*, one only needs to satisfy a single instance to validate NON-Primality.One can also obtain ALL the P by eliminating the BIM SIGO and O

^{2}from the 1^{st}Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O^{2}= ODDs^{2}, and the 1^{st}Diagonal = the 1^{st}Diagonal Parallel to the PD.This itself is further simplified by switching out the ODD AXIS values with the O

(13)^{2}— the O^{2}being the PD values — such that we now have:*SIGO(A*giving a distinct visualization advantage;^{2}) = Strict Inner Grid ODDS & ODD AXIS^{2}(14)

*NP = SIGO(A*^{2})(15)

*1st Diagonal - SIGO(A*.^{2}) = P

This second method — the algebraic geometry method — as presented here.

Two methods — one pure algebraic and the other a more visual algebraic geometry presented here — have been found that capture ALL the NO-PRIMES (NP). While they process slightly different, they dovetail nicely into a very visual Number Pattern Sequence (NPS) here on the BIM. They both give identical NP results.

So what is the significance of capturing ALL the NP?

The NP are the highly NPS that define the elusive pattern of the P. P + NP = ALL ODD WINs (≥3).

In any group of WIN, if you know the NP, you also know the P. Here is the highly visualizable geometric method for capturing ALL NP. In fact, it is just as simply stated in 12. of the SUMMARY.

One can obtain ALL the P by eliminating the BIM SIGO and O^{2} from the 1^{st} Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O^{2} = ODDs^{2}, and the 1^{st} Diagonal = the 1^{st} Diagonal Parallel to the PD.

This itself is further simplified by switching out the ODD AXIS values with the O^{2} — the O^{2} being the PD values — such that we now have:

One can also obtain ALL the P by eliminating the BIM SIGO and O^{2} from the 1^{st} Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O^{2} = ODDs^{2}, and the 1^{st} Diagonal = the 1^{st} Diagonal Parallel to the PD.

This itself is further simplified by switching out the ODD AXIS values with the O^{2} — the O^{2} being the PD values — such that we now have:

(14) *NP = SIGO(A ^{2})*

(15) *1st Diagonal - SIGO(A ^{2}) = P*.

**The PDFs will follow these animated gifs. Videos and other supporting graphics thereafter**.

__Animated Gifs:__

PRIMES vs NO-PRIMES-2: algebraic method.

PRIMES vs NO-PRIMES-1: algebraic-geometry method.

PRIMES vs NO-PRIMES-3: algebraic and algebraic geometry method.

PRIMES vs NO-PRIMES-4: algebraic method in detail.

__PDFs +__

PRIMES vs NO-PRIMES: snapshot-1 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below.

PRIMES vs NO-PRIMES PDF: snapshot-1 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The NPS of the NP define the elusive pattern of the P.

PRIMES vs NO-PRIMES: snapshot-2 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. Here the *x*=1,2,3,... base value sets -- Lower -*y* an Upper +*y*--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths. The larger PDF is below.

PRIMES vs NO-PRIMES PDF: snapshot-2 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the *x*=1,2,3,... base value sets -- Lower -*y* an Upper +*y*--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths.

PRIMES vs NO-PRIMES: snapshot-3 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. Here the *x*=1,2,3,... base value sets -- Lower -*y* an Upper +*y*--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths. The larger PDF is below.

PRIMES vs NO-PRIMES PDF: snapshot-3 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the *x*=1,2,3,... base value sets -- Lower -*y* an Upper +*y*--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths.

PRIMES vs NO-PRIMES PDF: snapshot-4 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the *x*=1,2,3,... base value sets -- Lower -*y* an Upper +*y*--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths.

Table31a4: PRIMES vs NO-PRIMES PDF: Open in a separate tab/window to see all 11 pages. Here the *x*=1,2,3,... base value sets -- Lower -*y* an Upper +*y*--are shown individually and collectively as sets. The full Upper and Lower tables conclude.

Table31a6_2: PRIMES vs NO-PRIMES PDF: If you look at the ODD Axis ÷3 NO-PRIMES (NP) that lie in the paths between the L-shaped Double-wide x-base sets (x=1,2,3,)…), one finds a distinct Number Pattern Sequence (NPS) between successive NP values. Those shown in BLUE are NOT included in the criss-crossing L-shaped Double-wide paths (seen in the snapshots above), while those in GRAY are. The NPS seen here, based ultimately on the 1,3,5,7,… ODD number summation series that defines the whole **BIM** distribution (including the 1st Diagonal, the PD, and the successive differences in sequential Inner Grid cell values) reiterates that of both the L-shaped Double-wide paths as well as the individual x paths. Both give a NPS of the NP that reveal the elusive pattern of the PRIMES. There remains little doubt that the PRIMES , as well as the Primitive Pythagorean Triples (PPTs), are intimately related to the **INVERSE SQUARE LAW (ISL)**!

__Videos:__

PRIMES vs NO-PRIMES-1-long from Reginald Brooks on Vimeo.

PRIMES vs NO-PRIMES-2-long from Reginald Brooks on Vimeo.

PRIMES vs NO_PRIMES 3 from Reginald Brooks on Vimeo.

PRIMES_vs_NO-PRIMES-4 from Reginald Brooks on Vimeo.

__...more supporting graphics and tables:__

Table31a4-5_+RunDiff_EQUATIONsDEMO(÷3Filter)+10x10+. A simple table to demo the NP process.

Table31a4_+RunDiff_EQUATION-PATTERNS-LOWER+UPPER+Annotated. NP Table with layout notes.

Table31a4_+RunDiff_EQUATIONs(SqrdAxis-seqPD)+50x500+.pdf. From the BIM to the NP/P Tables.

BIM35-Table31a4-5_NO-PRIMES_Factores-withARS-YELLOW-numbAnnotated. The NP pattern on the BIM.

Table33a,c,d_ODDsqrd÷24_sheets. The NP pattern – an amazing NPS – defines the P.

BIMrows1-1000+Primes_sheets+Sub-Matrix2. A BIM reference showing ÷24 ARS, PPTs, NP an P.

For the full documentation of all the graphics and pdfs — including history, development, expanded tables, and more — see Appendix B below.

While there are no specific references for this work other than referring back to my own original work, there are many references involved in the study and research on the PRIMES in general. These have been well documented in the TPISC_IV: Details_BIM+PPT+PRIMES focus white paper. This paper also contains the background graphics and tables leading up to this current work. The focus page is part of the much larger TPISC_IV: Details ebook project that is freely available as an HTML webpage. The quick reference outline can be found here.

REFERENCES:

https://primes.utm.edu/notes/faq/six.html

**“Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?**

"Perhaps the most rediscovered result about primes numbers is the following:

"I found that every prime number over 3 lies next to a number divisible by six. Using Matlab with the help of a friend, we wrote a program to test this theory and found that at least within the first 1,000,000 primes this holds true.

"Checking a million primes is certainly energetic, but it is not necessary (and just looking at examples can be misleading in mathematics). Here is how to prove your observation: take any integer *n* greater than 3, and divide it by 6. That is, write

*n* = 6*q* + r

where *q* is a non-negative integer and the remainder *r* is one of 0, 1, 2, 3, 4, or 5.

- If the remainder is 0, 2 or 4, then the number
*n*is divisible by 2, and can not be prime. - If the remainder is 3, then the number
*n*is divisible by 3, and can not be prime.

"So if *n* is prime, then the remainder *r* is either

- 1 (and
*n*= 6*q*+ 1 is one more than a multiple of six), or - 5 (and
*n*= 6*q*+ 5 = 6(*q*+1) - 1 is one less than a multiple of six).

"Remember that being one more or less than a multiple of six does not make a number prime. We have only shown that all primes other than 2 and 3 (which divide 6) have this form.”

From Another prime page by Chris K. Caldwell **caldwell@utm.edu**

~~~

"**Euler's 6n+1 Theorem**

"Every Prime of the form can be written in the form ."

https://archive.lib.msu.edu/crcmath/math/math/e/e282.htm

~~~

~~~

http://www2.mae.ufl.edu/~uhk/sixnplusone.pdf.

~~~

~~~

https://en.m.wikipedia.org/wiki/Formula_for_primes

~_{~}

http://www2.mae.ufledu/~uhk/PRIME-TEST.pdf ******

http://www2.mae.ufl.edu/~uhk/NUMBER- FRACTION.pdf

~~~

http://mathworld.wolfram.com/Eulers6nPlus1Theorem.html

~_{~}

http://eulerarchive.maa.org See Number Theory Section

E744en.pdf ******!!!!!!*****!!!!! MOST IMPORTANT READ

http://eulerarchive.maa.org/docs/translations/E744en.pdf

**NOTE: These article links from the Euler Archive/Number Theory/ INDEX are IMPORTANT READS for PRIMES & Squares. You can link directly from the Index as well.**

http://eulerarchive.maa.org/pages/E026.html

http://eulerarchive.maa.org/pages/E054.html

http://eulerarchive.maa.org/pages/E134.html

http://eulerarchive.maa.org/pages/E175.html

http://eulerarchive.maa.org/pages/E191.html

http://eulerarchive.maa.org/pages/E241.html

http://eulerarchive.maa.org/pages/E242.html

http://eulerarchive.maa.org/pages/E243.html

http://eulerarchive.maa.org/pages/E244.html

http://eulerarchive.maa.org/pages/E256.html

http://eulerarchive.maa.org/pages/E262.html

http://eulerarchive.maa.org/pages/E270.html

http://eulerarchive.maa.org/pages/E279.html

http://eulerarchive.maa.org/pages/E283.html

http://eulerarchive.maa.org/pages/E369.html

http://eulerarchive.maa.org/pages/E394.html

http://eulerarchive.maa.org/pages/E405.html

http://eulerarchive.maa.org/pages/E445.html

http://eulerarchive.maa.org/pages/E449.html

http://eulerarchive.maa.org/pages/E467.html

http://eulerarchive.maa.org/pages/E523.html

http://eulerarchive.maa.org/pages/E542.html

http://eulerarchive.maa.org/pages/E552.html

http://eulerarchive.maa.org/pages/E554.html

http://eulerarchive.maa.org/pages/E564.html

http://eulerarchive.maa.org/pages/E596.html

http://eulerarchive.maa.org/pages/E610.html

http://eulerarchive.maa.org/pages/E699.html

http://eulerarchive.maa.org/pages/E708.html

http://eulerarchive.maa.org/pages/E715.html

http://eulerarchive.maa.org/pages/E718.html

http://eulerarchive.maa.org/pages/E719.html

744 | On divisors of numbers of the form mxx + nyy |
---|---|

http://eulerarchive.maa.org original 744 Euler article

https://www.britannica.com/science/Fermats-theorem

http://mathworld.wolfram.com/FermatsLittleTheorem.html

http://mathworld.wolfram.com/EulersTotientTheorem.html

http://mathworld.wolfram.com/TotientFunction.html

https://www.geeksforgeeks.org/fermats-little-theorem/

https://www.geeksforgeeks.org/eulers-totient-function/

http://mathworld.wolfram.com/Eulers6nPlus1Theorem.html

https://brilliant.org/wiki/fermats-little-theorem/

https://brilliant.org/wiki/eulers-theorem/

https://primes.utm.edu/notes/proofs/FermatsLittleTheorem.html

https://primes.utm.edu/notes/conjectures/ Conjectures

https://primes.utm.edu/notes/faq/ FAQ Index

https://primes.utm.edu/notes/faq/one.html One

https://primes.utm.edu/notes/faq/six.html Six —>Important Page

http://unsolvedproblems.org/index.htm Unsolved Problems Index

https://en.m.wikipedia.org/wiki/Proofs_of_Fermat%27s_little_theorem

https://en.m.wikipedia.org/wiki/Fermat%27s_little_theorem

https://en.m.wikipedia.org/wiki/Euler%27s_theorem

~~~~~~~~~~NEW REFERENCE LIST 11-08-2018_{~}Fermat’s Little Theorem ++++Euler

https://web.math.princeton.edu/swim/SWIM%202010/Shi-Xie%20Presentation%20SWIM%202010.pdf

http://mathworld.wolfram.com/Fermats4nPlus1Theorem.html

http://nonagon.org/ExLibris/euler-proves-fermats-theorem-sum-two-squares

https://storyofmathematics.com/17th_fermat.html

https://en.wikipedia.org/wiki/Prime_number

http://mathworld.wolfram.com/PrimeNumberTheorem.html

https://www.britannica.com/science/prime-number-theorem

https://going-postal.com/2018/02/fermats-little-theorem/

https://ibmathsresources.com/2014/03/15/fermats-theorem-on-the-sum-of-two-squares/

https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares *****

https://en.wikipedia.org/wiki/Pythagorean_prime *****

https://oeis.org/A002144 Pythagorean Primes of the form 4n + 1

https://oeis.org/A002144/list List of Pythagorean Primes of the form 4n + 1

http://nonagon.org/ExLibris/fermat-sum-two-squares-calculator *****

https://www.britannica.com/science/Mersenne-prime Mersenne Primes

https://primes.utm.edu/mersenne/

https://en.wikipedia.org/wiki/Mersenne_prime#About_Mersenne_primes

http://mathworld.wolfram.com/MersenneNumber.html

http://mathworld.wolfram.com/MersennePrime.html

https://www.encyclopediaofmath.org/index.php/Mersenne_number

https://www.mersenne.org GIMPS

https://www.mersenne.org/primes/

https://primes.utm.edu/lists/small/millions/ Primes LIST by section

http://compoasso.free.fr/primelistweb/page/prime/liste_online_en.php Primes LIST by section

http://primerecords.dk/primegaps/gaps20.htm Prime Gaps

http://primerecords.dk/primegaps/maximal.htm

https://en.wikipedia.org/wiki/Prime_gap

https://primes.utm.edu/notes/GapsTable.html

https://en.wikipedia.org/wiki/Prime_k-tuple

https://primes.utm.edu/glossary/page.php?sort=ktuple

https://en.wikipedia.org/wiki/Twin_prime

http://mathworld.wolfram.com/TwinPrimes.html

https://primes.utm.edu/notes/faq/

~~

https://oeis.org/wiki/Pythagorean_primes Pythagorean vs non-Pythagorean Primes

http://oeis.org/A002145 non-Pythagorean Primes of the form 4n + 3

http://oeis.org/A002144 Pythagorean Primes of the form 4n + 1

https://en.wikipedia.org/wiki/Double_Mersenne_number Double Mersenne Primes

https://primes.utm.edu/mersenne/index.html#unknown Questions remain

~~~~~~~~~~NEW REFERENCE LIST 12-13-2018~~~videos——must watch~~~~

See: Animated math: videos by 3Blue1Brown (Grant Sanderson, et al.) https://www.3blue1brown.com/videos/

Feynman's Lost Lecture (ft. 3Blue1Brown) https://www.youtube.com/watch?v=xdIjYBtnvZU

But WHY is a sphere's surface area four times its shadow?: video by 3Blue1Brown https://youtu.be/GNcFjFmqEc8

Why is pi here? And why is it squared? A geometric answer to the Basel problem: video by 3Blue1Brown https://www.youtube.com/watch?v=d-o3eB9sfls&frags=pl%2Cwn

Pi hiding in prime regularities: video by 3Blue1Brown https://youtu.be/NaL_Cb42WyY

Visualizing the Riemann hypothesis and analytic continuation: video by 3Blue1Brown https://youtu.be/sD0NjbwqlYw

All possible pythagorean triples, visualized: video by 3Blue1Brown https://www.youtube.com/watch?v=QJYmyhnaaek

((((Table VI series is really the whole evolution of the BIMrow1-1000+sheets/Primes_sheets+PF, etc here and appendix.)))

referenced as Table VI a

referenced as Table VI b

referenced as Table VI b

Table VII.

BACK: ---> Part I of II CaCoST-DSEQEC on a separate White Paper
BACK: ---> Part II of II CaCoST-DSEQEC on a separate White Paper
BACK: ---> Simple Path BIM to PRIMES on a separate White Paper
BACK: ---> PRIMES vs NO-PRIMES on a separate White Paper
BACK: ---> TPISC_IV: Details_BIM+PTs+PRIMES on a separate White Paper
BACK: ---> PRIME GAPS on a separate White Paper

Artwork referencing TPISC.

Artwork referencing TPISC.

Here’s the thing. Amongst a myriad of other connections, there exist an intimate connection between three number systems:

- The
**ISL**as laid out in the**BIM**; - PTs — and most especially PPTs — as laid out on the
**BIM**; - The PRIME numbers — PRIMES — as laid out on the
**BIM**.

The **BIM** is the FIXED GRID numerical array of the **ISL**.

Amongst its vast array of inter-connecting **Number Pattern Sequences (NPS)** — i.e., number systems — two such systems stick out and do so in such an overtly visual — as well as mathematical — way that their connection to each other is more than implied.

You see, both the PPTs and PRIMES strictly align themselves on the SAME paths within the **BIM**.

Now, their footprints upon these paths are not identical, yet their paths chosen are. If you divide the number array — i.e., the Inner Grid numbers — of the **BIM** by 24, a **Sub-Matrix 1** grid is formed.

Upon that Sub-Matrix 1, pathways are formed on every ODD Axis number NOT ➗3.** EVERY PPT and PRIME lies on these paths!**

Yes, while in the details we show how:

- The 1st Col. on the
**BIM**— that which is the square of the Axis number (i.e., the Prime Diagonal number) - 1, when then ➗ by 24, equals a**Whole Integer Number (WIN)**; - This defines the path — the “
**Active Row**” upon the**BIM**; - While every
**Active Row**path may or may not contain a PPT and/or PRIME, every PPT and/or PRIME is ALWAYS located on one of these paths; - The difference (∆) between the squares of any two PRIMES (>5) is also ➗24;
- The serial — and exponential — products of ANY and ALL PPTs remain PPTs, whilst those NOT remain NOT.

Furthermore, by distilling the **BIM** to **Sub-Matrix 2** — i.e., every number across a Row is progressively divided by a growing sequence series starting with the Col. 1 as the Axis number - 1 — every such serial-exponential PPT is clearly predicted by its neighbors within the sequence. One more example of the extremely intimate relationship between **The Pythagorean - Inverse Square Connection (TPISC)**.

One simply can not ignore how the PRIMES, the PTs and ultimately the ISL define the architecture of SpaceTime!

In the original *MathspeedST*, an artificial division was made, separating the content into:

- TAOST, The Architecture Of SpaceTime,
- TCAOP, The Complete Absence Of Primes.

Now we have come full circle.

ALL PRIMES and ALL PPTs follow — although individually with their own respective footprints — the SAME, HIGHLY PATTERNED **NPS** path on the **BIM**.

This is no coincidence. The ➗24 **Active** **Row** Pattern that defines this path on the **BIM** does so in a highly ordered pattern. The energy density that expresses itself as curved ST does so precisely by the numerical architect of the built-in **ISL**.

We now have some very strong evidence that the numbers that define the PPTs and the numbers that define the PRIMES are ubiquitously linked throughout the entire number pattern array that defines the **ISL**. This is revealed on the **BIM**!

Surely their interplay provides some very significant contributions to the overall Architecture of SpaceTime!

VIMEO "BIM_PRIMES_24"1! )

BIM_PT_PRIMES_24 from Reginald Brooks on Vimeo.

BACK: ---> Part I of II CaCoST-DSEQEC on a separate White Paper
BACK: ---> Part II of II CaCoST-DSEQEC on a separate White Paper
BACK: ---> TPISC_IV: Details:_PRIMES_vs_NO-PRIMES on a separate White Paper
BACK: ---> Simple Path BIM to PRIMES on a separate White Paper
BACK: ---> PRIMES vs NO-PRIMES on a separate White Paper
BACK: ---> TPISC_IV: Details_BIM+PTs+PRIMES on a separate White Paper
BACK: ---> PRIME GAPS on a separate White Paper
BACK: ---> TPISC IV: Details White Paper

Artwork referencing TPISC.

Artwork referencing TPISC.

~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~

**SEE Page 50 (Art Theory101: White Papers) for spreadsheets:** BIM: How to Make.

TABLES 1-9 (r-sets, Z➗24,Axis squared, differences, groundwork tables)

(Once renamed, ALL Tables — as pdf/pngs—will be listed here with links to and fro main article)

TABLES 10-19 (PRIMEs-gaps and random differences)

TABLES 20-23 (PRIME-PPT or not with r-values)

TABLES 24-28 (Sub-Matrix 2 )

TABLES 29-30 (*Sub-Matrix 2 Sidebar*: Exponentials of the PPTs)

**Table 29**Exponentials of the first 10 PPTs-values to be used in*c***Tables 30a-g**.**Tables 30a-g**The**Sub-Matrix 2**, when ➗4, and the difference (∆) between this and the next exponential PPT treated this way, is subsequently ➗ by its**Sub-Matrix 2**__variable__, the PREVIOUS exponential within the series is revealed. Restated as an example: When one subtracts 1 from the exponential values of(the*c*-value of the PPT) you get the*c***Sub-Matrix 2**value. Divide that by 4 and take the difference (∆) between it and the next. Divide that by 3 to give the PREVIOUS PPT-value in the series.*c*The

**Sub-Matrix 2**variable divisor = 3 =**Sub-Matrix 2**value/4 = 12/4. These variables run: 1,3,4,6,7,9,10,...

Tables: 31a, a+, a++, ++MP, a+++ and 31b

Tables: 32a,32b and 32c

**SEE: Tables: 33a, 33b, 33c and 33d. towards the end of Appendix B for some very NEW INFO on the BIM÷24.**

The underlying geometry of the

The

The interplay between these small sets of Numbers generates an incredible amount of richness and complexity with seemingly simplistic

Open in separate browser tab/window to see all.

Supporting Graphics forTables: 33a, 33b, 33c and 33d. Open in separate browser tab/window to see all.

xxxxxxxxBack to the Table 31 sequences. Table 31a... series, along with the previous Table33 series, uncovers how the PRIMES were found within the **BIM** both by algebra and geometry. This has been reduced to a focus white paper: "** PRIMES vs NO-PRIMES** that is a condensed summary of this work. One may benefit from looking over this paper prior to the study of these more elaborate tables. A summary from the paper is presented here:

The BIM is a symmetrical grid — divided equally down its diagonal center with the Prime Diagonal (PD) — that illuminates the Number Pattern Sequence (NPS) of the Inverse Square Law (ISL) via simple, natural Whole Integer Numbers (WIN).

The BIM Axis numbers are 1,2,3,.. with 0 at the origin.

The Inner Grid (IG) contains EVEN and ODD WIN, but except for the 1st diagonal next to the PD — a diagonal that contains ALL the ODD WIN — there are NO PRIMES (NO-PRIMES, NP) on the SIG (Strict Inner Grid).

The PD WIN are simple the square of the Axis WIN.

ALL the IG WIN result from subtracting the horizontal from the vertical intersection of the PD.

Dropping down a given PD Squared WIN (>4) until it intersects with another squared WIN on a Row below will ALWAYS reveal that Row to be a Primitive Pythagorean Triple (PPT) Row, whose hypotenuse,

*c*, lies on the intersecting PD. ALL PPTs may be identified this way.Dividing the BIM cell values by 24 — BIM÷24 — forms a criss-crossing DIAMOND NPS that divides the overall BIM into two distinct and alternating Row (and Column) bands or sets:

- ODD WIN that are ÷3 and referred to as NON-ARs;
- ODD WIN that are NOT ÷3 and referred to as ARs, or Active Rows;
- The ARs ALWAYS come in pairs — with an EVEN WIN between — as the UPPER and LOWER AR of the ARS (Active Row Set);

ALL PPTs and ALL PRIMES ALWAYS are found exclusively on the ARs — no exceptions.

By applying:

(1) *

let

*y*= odd number (ODD) 3, 5, 7,… and*x*= 1, 2, 3,... one generates a NP table containing ALL the NP;*True if ÷3 ODDs are first eliminated, otherwise ADD exponentials of 3 to the NP pool;

Eliminating the NP — and the NP contain a NPS — from ALL the ODD WIN, reveals the PRIMES (P).

A necessary, but not sufficient confirmation — but not proof — of primality is found by finding the even division of 24 into the difference of the square of ANY two PRIMES as:

(2) (P

_{2})^{2}- (P_{1})^{2}=let

Be aware that this also holds true for ALL the AR NP. The P and NP are NOT ÷3, and are both part of the ARS and therefore any combination of the two squared differences will be ÷24:

(3) (NP

_{2})^{2}- (NP_{1})^{2}=(4) (NP

_{2})^{2}- (P_{1})^{2}=(5) (P

_{2})^{2}- (NP_{1})^{2}=The ÷3 NON-AR set is separately ÷24, but can NOT be mixed with members of the AR set (ARS) as:

(6) (NON-AR-NP

_{2})^{2}- (NON-AR-NP_{1})^{2}= n24(7) (NON-AR-NP

_{2})^{2}- (NP_{1})^{2}⧣(8) (NP

_{2})^{2}- (NON-AR-NP_{1})^{2}⧣(9) (P

_{2})^{2}- (NON-AR-NP_{1})^{2}⧣(10) (NON-AR-NP

_{2})^{2}- (P_{1})^{2}⧣The division into AR and NON-AR sets has a NPS that ultimately define the elusive pattern of the P.

Furthermore, may be re-arranged to:

(11)

(12)

asking whether any given ODD (>3) is a P or NP, it is exclusively a NP if, and only if,

*y*reduces to the same value after applying*x*. As*y*is effectively an ODD of either a PRIME or composite of PRIMES factor*, one only needs to satisfy a single instance to validate NON-Primality.One can also obtain ALL the P by eliminating the BIM SIGO and O

^{2}from the 1^{st}Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O^{2}= ODDs^{2}, and the 1^{st}Diagonal = the 1^{st}Diagonal Parallel to the PD.^{2}— the O^{2}being the PD values — such that we now have:*SIGO(A*giving a distinct visualization advantage;^{2}) = Strict Inner Grid ODDS & ODD AXIS^{2}(14)

*NP = SIGO(A*^{2})(15)

*1st Diagonal - SIGO(A*.^{2}) = P

This second method — the algebraic geometry method — as presented here.

Two methods — one pure algebraic and the other a more visual algebraic geometry presented here — have been found that capture ALL the NO-PRIMES (NP). While they process slightly different, they dovetail nicely into a very visual Number Pattern Sequence (NPS) here on the BIM. They both give identical NP results.

So what is the significance of capturing ALL the NP?

The NP are the highly NPS that define the elusive pattern of the P. P + NP = ALL ODD WINs (≥3).

In any group of WIN, if you know the NP, you also know the P. Here is the highly visualizable geometric method for capturing ALL NP. In fact, it is just as simply stated in 12. of the SUMMARY.

One can obtain ALL the P by eliminating the BIM SIGO and O^{2} from the 1^{st} Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O^{2} = ODDs^{2}, and the 1^{st} Diagonal = the 1^{st} Diagonal Parallel to the PD.

^{2} — the O^{2} being the PD values — such that we now have:

^{2} from the 1^{st} Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O^{2} = ODDs^{2}, and the 1^{st} Diagonal = the 1^{st} Diagonal Parallel to the PD.

^{2} — the O^{2} being the PD values — such that we now have:

(14) *NP = SIGO(A ^{2})*

(15) *1st Diagonal - SIGO(A ^{2}) = P*.

**The PDFs will follow these animated gifs. Videos and other supporting graphics thereafter**.

__Animated Gifs:__

PRIMES vs NO-PRIMES-2: algebraic method.

PRIMES vs NO-PRIMES-1: algebraic-geometry method.

PRIMES vs NO-PRIMES-3: algebraic and algebraic geometry method.

PRIMES vs NO-PRIMES-4: algebraic method in detail.

__PDFs +__

PRIMES vs NO-PRIMES: snapshot-1 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below.

PRIMES vs NO-PRIMES PDF: snapshot-1 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The NPS of the NP define the elusive pattern of the P.

PRIMES vs NO-PRIMES: snapshot-2 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. Here the *x*=1,2,3,... base value sets -- Lower -*y* an Upper +*y*--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths. The larger PDF is below.

PRIMES vs NO-PRIMES PDF: snapshot-2 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the *x*=1,2,3,... base value sets -- Lower -*y* an Upper +*y*--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths.

PRIMES vs NO-PRIMES: snapshot-3 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. Here the *x*=1,2,3,... base value sets -- Lower -*y* an Upper +*y*--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths. The larger PDF is below.

PRIMES vs NO-PRIMES PDF: snapshot-3 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the *x*=1,2,3,... base value sets -- Lower -*y* an Upper +*y*--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths.

PRIMES vs NO-PRIMES PDF: snapshot-4 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the *x*=1,2,3,... base value sets -- Lower -*y* an Upper +*y*--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths.

Table31a4: PRIMES vs NO-PRIMES PDF: Open in a separate tab/window to see all 11 pages. Here the *x*=1,2,3,... base value sets -- Lower -*y* an Upper +*y*--are shown individually and collectively as sets. The full Upper and Lower tables conclude.

Table31a6_2: PRIMES vs NO-PRIMES PDF: If you look at the ODD Axis ÷3 NO-PRIMES (NP) that lie in the paths between the L-shaped Double-wide x-base sets (x=1,2,3,)…), one finds a distinct Number Pattern Sequence (NPS) between successive NP values. Those shown in BLUE are NOT included in the criss-crossing L-shaped Double-wide paths (seen in the snapshots above), while those in GRAY are. The NPS seen here, based ultimately on the 1,3,5,7,… ODD number summation series that defines the whole **BIM** distribution (including the 1st Diagonal, the PD, and the successive differences in sequential Inner Grid cell values) reiterates that of both the L-shaped Double-wide paths as well as the individual x paths. Both give a NPS of the NP that reveal the elusive pattern of the PRIMES. There remains little doubt that the PRIMES , as well as the Primitive Pythagorean Triples (PPTs), are intimately related to the **INVERSE SQUARE LAW (ISL)**!

__Videos:__

PRIMES vs NO-PRIMES-1-long from Reginald Brooks on Vimeo.

PRIMES vs NO-PRIMES-2-long from Reginald Brooks on Vimeo.

PRIMES vs NO_PRIMES 3 from Reginald Brooks on Vimeo.

PRIMES_vs_NO-PRIMES-4 from Reginald Brooks on Vimeo.

__...more supporting graphics and tables:__

Table31a4-5_+RunDiff_EQUATIONsDEMO(÷3Filter)+10x10+. A simple table to demo the NP process.

Table31a4_+RunDiff_EQUATION-PATTERNS-LOWER+UPPER+Annotated. NP Table with layout notes.

Table31a4_+RunDiff_EQUATIONs(SqrdAxis-seqPD)+50x500+.pdf. From the BIM to the NP/P Tables.

BIM35-Table31a4-5_NO-PRIMES_Factores-withARS-YELLOW-numbAnnotated. The NP pattern on the BIM.

Table33a,c,d_ODDsqrd÷24_sheets. The NP pattern – an amazing NPS – defines the P.

BIMrows1-1000+Primes_sheets+Sub-Matrix2. A BIM reference showing ÷24 ARS, PPTs, NP an P.

BIM250x250.

Various supporting graphics

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BACK: ---> Part I of II CaCoST-DSEQEC on a separate White Paper
BACK: ---> Part II of II CaCoST-DSEQEC on a separate White Paper
BACK: ---> Simple Path BIM to PRIMES on a separate White Paper
BACK: ---> PRIMES vs NO-PRIMES on a separate White Paper
BACK: ---> TPISC_IV: Details_BIM+PTs+PRIMES on a separate White Paper
BACK: ---> PRIME GAPS on a separate White Paper
BACK: ---> PeriodicTableOfPrimes(PTOP)_GoldbachConjecture on a separate White Paper
NEWLY ADDED (after TPISC IV published):

Artwork referencing TPISC.

Back to Part I of the BIM-Goldbach_Conjecture.

Back to Part II of the BIM-Goldbach_Conjecture.

Back to Part III of the BIM-Goldbach_Conjecture.

Artwork referencing TPISC.

Artist Link in iTunes Apple Books Store: Reginald Brooks

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Art Theory 101 / White Papers Index

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