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. 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 visable geometric method for capturing ALL NP. In fact, it is just as simply stated in the SUMMARY below.
Algebraic Geometry Method
One can obtain ALL the P by eliminating the BIM SIGO and O^2 from the 1st Diagonal WIN,
where SIGO = Strict Inner Grid ODDs, O^2 = ODDs^2, and the 1st Diagonal = the 1st 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:
▪ SIGO(A^2) = Strict Inner Grid ODDS & ODD AXIS^2 giving a distinct visualization advantage;
▪ NP = SIGO(A^2)
▪ 1st Diagonal - SIGO(A^2) = P.
(~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 theNumber 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
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, 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.
…more at: brooksdesign-ps.net/Reginald_Brooks/Code/Html/MSST/MSST-TPISC_resources/MSST-TPISC_resources.html#TPISC_IV:%20Details
Thanks for viewing.