PRIMES vs NO-PRIMES

A definitive method revealing the pattern of the primes.

Copyright © 2019, Reginald Brooks, Brooks Design. All rights reserved.

ABSTRACT

Identifying the PRIMES (P) from the NON-PRIMES or NO-PRIMES (NP) from the pool of ODD numbers is a matter of separation, as one defines the other. Two new methods have been found: the algebraic method and the algebraic-geometry method. Both methods capture ALL the NP and visually come together in the geometry. The focus here is on the algebraic method.

The equation, 6yx +/- y = NP, will identify ALL NP*. Eliminating these NP from the the ODD number pool will leave only the P. A confirmation of primality (but not proof) is found simply be showing the difference in the squares of ANY two P(>3) is a multiple of 24.

*True if ÷3 ODDs are first eliminated, otherwise ADD exponentials of 3 to the NP pool.

Any ODD number can be proved to a a NP if, and only if, its ODD factor (y) divides evenly said ODD/6, as reduced by x, where x=1,2,3,..

A distinct Number Pattern Sequence (NPS) of the NP defines the elusive pattern of the P.

INTRODUCTION and PREMISE

Identifying the PRIMES (P) from the NON-PRIMES or NO-PRIMES (NP) from the pool of ODD numbers is a matter of separation — as one defines the other. Equation (1) will identify ALL NP.

      (1)                  * $6yx ± y = NP$

let y = odd number (ODD) 3, 5, 7,… and x = 1, 2, 3,...

*True if ÷3 ODDs are first eliminated, otherwise ADD exponentials of 3 to the NP pool.

It is advantageous to initially remove ALL ODDs ÷3 — and similarly from the NP÷3 — as this reduces the pool by a third. The remaining NP set = AR, an Active Row, while the ÷3 ODDs = NON-AR set.

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)                  (P₂)² - (P₁)² = $n24$

let $n = 1, 2, 3,...$

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)                  (NP₂)² - (NP₁)² = $n24$

      (4)                  (NP₂)² - (P₁)² = $n24$

      (5)                  (P₂)² - (NP₁)² = $n24$

The ÷3 NON-AR set is separately ÷24, but can NOT be mixed with members of the AR set (ARS) as:

      (6)                  (NON-AR-NP₂)² - (NON-AR-NP₁)² = n24

      (7)                  (NON-AR-NP₂)² - (NP₁)² ⧣ $n24$

      (8)                  (NP₂)² - (NON-AR-NP₁)² ⧣ $n24$

      (9)                  (P₂)² - (NON-AR-NP₁)² ⧣ $n24$

      (10)                (NON-AR-NP₂)² - (P₁)² ⧣ $n24$

The division into AR and NON-AR sets has a NPS that ultimately define the elusive pattern of the P.

Furthermore, $6yx ± y = NP$ may be re-arranged to:

      (11)                  $(NP ± y)/6yx = 0$

      (12)                  $(NP ± y)/6x = y$

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.

If, and only if, $(NP ± y) )/6x = y$ is true, then that ODD number is a NP.

*The various instances includes composites of the given prime factors, but only the pure prime factors need to be satisfied.

CONCLUSION

The NP form a distinct NPS, and in doing so, they draw out the opposing — and elusive — pattern of the P.

A positive and negative space collectively forming an image shape is apparent.

The philosophical paradox questions which is the positive and which is the negative space.

As in the matter-antimatter duality, our preferred frame of reference limits us from seeing — and likely fully experiencing — the whole.

As ALL NP can be easily surmised via a spreadsheet, eliminating these from the pool of ODDs surveyed, will reveal ALL the P. This may be done for any contiguous run of ODDs. This inclusive-exclusive approach will not only predict all larger P, it will, indeed, reveal the possibility of ALL hidden primes between — as with those between the larger Mersenne Primes.

BACKGROUND

The full details of the research and discoveries are documented here.

• A relationship between the P and 24 was found in 2005 (link)

• Years later, a right-angle, squared numbers relationship between the ODD number summation series and the Inverse Square Law (ISL) lead to the discovery of the BIM (BBS-ISL Matrix).

• The BIM is infinite and ubiquitous in presenting the known relationship of natural Whole Integer Numbers (WIN) in describing the Universal Law of the ISL. It is easily laid out in a spreadsheet app.

• The Pythagorean Triples — Pythagorean Triangles composed exclusively of WIN — are also infinitely and ubiquitously embedded within the BIM. There is a connection of the Primitive PTs (PPTs) to the P.

• TPISC I, II, III, IV (The Pythagorean–Inverse Square Connection) fully describes the Basic, Advanced, Clarity and the Tree of Pythogorean Triples, and Details of the PPTs and PRIMES, as well as the Double-Slit Experiment and Quantum Entanglement Conjecture (DSEQEC), as part of the larger Creation and Conservation of SpaceTime Conjecture (CaCoST).

• Dividing the WIN within the BIM by 24 — BIM÷24 — has led to the discovery:

  • ALL PPTs and ALL PRIMES satisfy a necessary, but not sufficient, condition that they ONLY fall on NOT ÷3 ODD numbered Rows on the BIM;

  • Such Row are referred to as Active Rows (ARs) within an ODD–EVEN–ODD Row set, the Active Row Set (ARS);

  • ALL ÷3 Row — NON-ARs — alternate with the ARs on the BIM;

  • The NPS of ALL NP:

    • those ÷3 = NON-AR
    • those NOT ÷3 = AR

    reveal ALL NP and in doing so, by elimination, reveal ALL P. See supporting graphics and tables below;

  • Eliminating ALL ODDs ÷3 removes ALL NP÷3, simplifying the base of ODDs to ONLY those on ARS;

  • $6yx ± y = NP$ identifies ALL NP*;

    *True if ÷3 ODDs are first eliminated, otherwise ADD exponentials of 3 to the NP pool.;

    
    

  • PRIMES (P) are ALL that remain once the NP are eliminated;

  • $n24$ divides EVENLY into:

    • ALL differences between ANY two squared ARs — includes both P and NOT÷3 NP;
    • ALL differences between ANY two squared NON-ARs — includes exclusively the ÷3NP;
    • NO cross-over between the two sets allowed.

  • Re-arranging $6yx ± y = NP$ , one may ask if any ODD number is P or NP;

    • ODD = NP if, and only if, the y factor reduces to the same value upon dividing the ODD ±y by 6x;
    • ODD = PRIME if, and only if, no y factor reduces to the same value.

SUMMARY*

(~from "*Simple Path from the BIM to the PRIMES" that presents the algebraic-geometry method.)

How do we go from the simple grid of the BIM (BBS-ISL Matrix) to identifying the PRIMES?

1. 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).

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

3. 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).

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

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

6. 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.

7. 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);

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

9. By applying:

         (1)                  * $6yx ± y = NP$

   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;

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

11. 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)                  (P₂)² - (P₁)² = $n24$

    let $n = 1, 2, 3,...$

    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)                  (NP₂)² - (NP₁)² = $n24$

          (4)                  (NP₂)² - (P₁)² = $n24$

          (5)                  (P₂)² - (NP₁)² = $n24$

    The ÷3 NON-AR set is separately ÷24, but can NOT be mixed with members of the AR set (ARS) as:

          (6)                  (NON-AR-NP₂)² - (NON-AR-NP₁)² = n24

          (7)                  (NON-AR-NP₂)² - (NP₁)² ⧣ $n24$

          (8)                  (NP₂)² - (NON-AR-NP₁)² ⧣ $n24$

          (9)                  (P₂)² - (NON-AR-NP₁)² ⧣ $n24$

          (10)                (NON-AR-NP₂)² - (P₁)² ⧣ $n24$

    The division into AR and NON-AR sets has a NPS that ultimately define the elusive pattern of the P.

    Furthermore, $6yx ± y = NP$ may be re-arranged to:

          (11)                  $(NP ± y)/6yx = 0$

          (12)                  $(NP ± y)/6x = y$

    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.

12. One can also obtain ALL the P by eliminating the BIM SIGO and O² from the 1^st Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O² = ODDs², 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 O² — the O² being the PD values — such that we now have:

          (13)   SIGO(A²) = Strict Inner Grid ODDS & ODD AXIS² giving a distinct visualization advantage;

          (14)                  NP = SIGO(A²)

          (15)                  1st Diagonal - SIGO(A²) = P.

This second method — the algebraic geometry method — is presented here. (Simple Path from the BIM to the PRIMES.)

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: ---> 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

BACK: ---> Make the PTOP with Fractals on a separate White Paper

BACK: ---> TPISC IV: Details White Paper

Artwork referencing TPISC.