TPISC_IV:_Details_BIM+PTs+PRIMES

Preface

C. How — but NOT why — 24 connects the PTs and PRIMES to the BIM.

(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.)

Sub-Matrix

(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.

Together Sub-Matrix 1 and Sub-Matrix 2 will provide the visual-geometric and algebraic-geometric location of ALL PPTs and PRIMES.

First Sub-Matrix 1:

1. 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.

2. 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.

3. 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_f') and it, too, always and only exists on a YELLOW marked Row (“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 4A (A=area) value of that PT and the “step-sister” PPT Row always contains the 8A 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!

• 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 = a² and b²) and the 4A value ONLY, it is a PPT Row.

  If it contains BOTH the 4A and 8A values, it is a 'step-sister' Row, i.e. (c²-1)/24 and (c²-25)/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.

  (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 (c²) 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,... )

• 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:

  1. ALL PTs (gray with small black dot) fall on an ACTIVE Row.

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

  3. The difference, ∆, in the SQUARED Axis #s on any two ACTIVE Rows is ALWAYS divisible by 24.

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

  5. The difference, ∆, in the SQUARED Axis #s on any non-ACTIVE ODD Row and another non-ACTIVE ODD Row is ALWAYS divisible by 24.

  6. Going sequentially down the Axis, every ODD number in the series follows this pattern:

     nA—A-A—nA—A-A—nA—A-A—

     ​

  7. #3—5-7—9—11-13—15—17-19—21... Every 3rd ODD # (starting with 3) is ➗by 3 = nA .

  8. #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.

  9. #3,4,5 re-stated: let A = ACTIVE Row Axis #, nA = non-ACTIVE Row Axis #

     A₂²-A₁²= ➗ 24 and A ≠ ➗by 3

     nA²-A²≠ ➗ 24

     nA₂²-nA₁²= ➗ 24 and nA = ➗by 3

  10. 6,7,8 re-stated:

      ODD Axis #s ➗by 3 (every 3rd ODD #) are NEVER ACTIVE Row members — thus never PT/PRIME

      ODD 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 # :

  ​ A₁²+ 24x = A₂² and √A₂² = A₂ = a PT and/or PRIME candidate;

  ​ ODD₁²+ 24x = ODD₂² and √ODD₂² = ODD₂ = 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).

  ​

  ​

  BIM➗PPTs and PRIMES: (Latest: as this work was being prepared, a NEW relationship was found.) See below under 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 BIM (Tables 24-28).

 

 

 

 

Now, we introduce

Sub-Matrix 2:

• 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:

1. Axis# must be ODD, NOT ➗3 = Active Row Set (ARS) member;

2. Only 1 of the 2 ARS can be a PPT;

3. Sub-Matrix Col 1 # MUST be ➗4;

4. SOME may NOT be PPT if ➗Prime Factor (>5);

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.

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Sub-Matrix 2 Sidebar: Exponentials of the PPTs

• Table 29 Exponentials of the first 10 PPTs c-values to be used in 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 c (the c-value of the PPT) you get the 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 c-value in the series.

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

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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 (x²-1)/24 is true;

• 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!

 

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BIM÷24: SubMatrix Sidebar: What is the role of 24 in the underlying structure?

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 BIM÷24 PRE-SELECTS the Axis Rows into TWO Groups: ARs and NON-ARs. The PPTs and PRIMES are EXCLUSIVELY — as a sufficient, but not necessary condition — found on the ARs and NEVER on the NON-ARs. While both Groups follow (PD² - PD²)÷24, they do so ONLY within their own respective Groups. They do NOT crossover. This Grouping divide occurs naturally within the BIM as shown in these images below.

The ISL as presented in the BIM is deeply, intimately structured around the number 24 — and its factors: 4,6, 3,8 2,12, and 1,24.

The interplay between these small sets of Numbers generates an incredible amount of richness and complexity with seemingly simplistic BIM itself. This has led to TPISP: The Pythagorean-Inverse Square Connection, and the PRIMES.

Open in separate browser tab/window to see all.

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BIM➗PPTs and PRIMES

BIM➗PPTs and PRIMES: (Latest: as this work was being prepared, a NEW relationship was found.)

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 ARs 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 ARs may be a PPT, while both, neither, or one or the other ARs 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 ARs. 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

  741²-5²/24 = n= 22877.3 NOT PRIME

• 189

  189²-5²/24 = n = 1487.3 NOT PRIME

• 289

  289²-5²/24 = n = 3479 PRIME

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  If P = X, evenly ➗24, then P = a PRIME Candidate where, X=evenly➗24, P=Prime, P²₂=larger Prime Squared, P²₁=smaller PRIME Squared, n=1,2,3,…

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