PTOP

For every EVEN # that, when subtracting from it 3 (P₁), gives a difference (∆) of another P, a 3-PRIME PPset — designated as “3, P₂” or “3 + P₂”, the sum (∑) P₁ + P₂ = EVEN # — a PPset Trail is formed.

The PPset Trail follows a strict Number Pattern Sequence (NPS):

• Periodic Table of PRIMES (PTOP) has Column Headers of the PRIMES Sequence 3,5,7,11,13,17,19,23,….;

• The PPset Trail starts on the PTOP under the PRIME = 3 Column;

• It ONLY starts for EVENs formed from “3 + P₂” sets;

• The P₂ itself sequentially increases DOWN the Column with the SAME NPS spacing as the PRIMES Sequence — leaving a blank space for each PRIME Gap in the sequence;

• The PPset Trail ALWAYS follows the SAME IDENTICAL NPS as it forms a jagged diagonal from Top Left to Bottom Right;

• The PPset Trail NPS is the SAME PRIMES Sequence 3,5,7,11,13,17,19,23,…, but BEGINS the sequence at that PRIME at which that EVEN # - 3 = that PRIME;

• The PPset Trail NPS going sequentially down the diagonal, will skip the next diagonal space – leave blank – for the PRIME Gaps;

• For those EVENs that are NOT “3 + P₂” sets, the remaining Columns of the PRIMES Sequence 5,7,11,13,17,19,23,…. will fill in, picking up on a previous PPset Trail from a smaller EVEN above;

• For those EVENS that are NOT “3 + P₂” sets, the PPset Trail NPS remains the SAME and IDENTICAL as the “3 + P₂” sets, the ONLY difference is that it starts further down the Trail;

• The diagonal length of the PPset Trail increases by 1 PRIME # for each successive “3 + P₂” sets and that ∑# of PPsets is given in the 1^st Header Column;

• The # of EVENS Covered (EC) by a given PPset Trail — the Ending EVEN # (E_e) - Starting EVEN # (E_s) = EC — is important in establishing further NPS as seen in Tables 38-40.

• The E_e = 2P₂ (see Tables 38-40);

• Going horizontally across the PTOP, those PPsets from the various diagonally crossing PPset Trails will add up to the ∑ of PPsets that form that EVEN #, as shown in the last Column;

• While it is difficult to visualize here, or in the Tables 38-40, on the BIM one can readily see that EACH PPset is the base of a 90° R-angled isosceles triangle with its apex pointing to the EVEN #! (See rule170annotated image.)

The combination of two (2) PRIME Sequences, P₁ and P₂ — bifurcating at the P₂ — not only gives one or more PPsets for every EVEN#, it also reveals a starling NEW NPS for the PRIMES: PRIMES + PRIMES generates 90° R-angled isosceles triangles!

PTOP: Periodic Table Of PRIMES & the

Goldbach Conjecture

INTRODUCTION

While the (strong) Goldbach Conjecture has been verified up to 4x10¹⁸, it remains unproven.

A number of attempts have demonstrated substantial, provocative and often beautiful patterns and graphics, none have proven the conjecture.

Proof of the conjecture must not rely solely on the notion that extension of a pattern to infinity will automatically remain valid.

No, instead, a proof must, in its very nature, reveal something new about the distribution and behavior of PRIMES that it is absolutely inevitable that such pattern extension will automatically remain valid. The proof is in the pudding!

Proof offered herein is just such a proof. It offers very new insights, graphical tables and algebraic geometry visualizations into the distribution and behavior of PRIMES.

In doing so, the Proof of the Euler Strong form of the Goldbach Conjecture becomes a natural outcome of revealing the stealthy hidden Number Pattern Sequence (NPS) of the PRIMES.

STATEMENT: Layout & Essentials

Proof of the Goldbach Conjecture (strong form, ≥6)

1. Natural (n), Whole Integer Numbers (WIN) — 0,1,2,3,…infinity — form horizontal and vertical Axis of a simple matrix grid.

2. The squares of such WINs — n²=1²=1, 2²=4, 3²=9,…infinity — forms the central Diagonal of said grid — dividing it into two bilaterally symmetric triangular halves.

3. Every Inner Grid (IG) cell within is simply the difference (∆) between its horizontal and vertical Diagonal intercept values. They extend to infinity. The Diagonal WINs form the base of a 90° R-angled isosceles triangle with said IG cell value at the apex.

4. Every IG cell within is also the product of two Axis WINs (Either horizontal or vertical, not both), that form the base of a 90° R-angled isosceles triangle with said IG cell value at the apex.

5. The complete matrix grid extends to infinity and is referred to as the BIM (BBS-ISL Matrix).The BIM forms — and informs — a ubiquitous map (algebraic geometry) to:

   ​ The Inverse Square Law (ISL);

   • The Pythagorean Triples (PT);
   • The PRIMES (stealthily hidden, but revealed by NPS.

6. The 1^st Diagonal that runs parallel to either side of the main Prime Diagonal (PD, not of PRIME numbers, but primary), is composed of the ODD WINs: 1,3,5,…infinity.

7. If we add +1 to each value, that 1^st Diagonal now becomes a sequence of ALL the EVEN WINs (≥4): 4, 6, 8,…infinity. NOTE: this is why the PTOP is hidden, in the normal, base BIM these remain ODDs.

8. Select ANY EVEN WIN and plot a line straight back to its Axis WIN — that Axis WIN = EVEN/2 = core Axis #.

9. Upon that same Axis, PRIME Pair sets (PPsets) — whose sum (∑) equals the EVEN WIN (on the 1^st Diagonal) — will be found that form the base of 90° R-angled isosceles triangle(s) whose apex lie(s) on that straight line between the EVEN and its 1/2 Axis WINs. PPsets with identical PRIMES = 1/2 Axis value.

10. The proof that every EVEN WIN has ≥1 PPsets can be seen in the Periodic Table Of PRIMES (PTOP) that stealthily informs the BIM of how each and every EVEN WIN is geometrically related to one or more PPsets.

11. These PPsets are NOT randomly contributing their ∑s to equal the EVEN WINs, rather they come about as the consequence of a strict NPS: the sequential — combining, linking, concatenation — addition of the PRIMES Sequence (PS) — 3,5,7,11,13,17,19,…—to a base PS — 3,5,7,11,13,17,19,….

12. The NPS of this addition forms the PTOP: for each vertical PS — the 1^st PRIME (P₁) remains constant (3), the 2^nd PRIME (P₂) sequentially advances one (1) PS WIN — is matched diagonally with the 2^nd PS, but now the 1^st PRIME sequentially advances, while the 2^nd PRIME remains constant within a given PPset.

13. This matching addition of the 2^nd PS at the bifurcation point of the common 2^nd PRIMES, forms the zig-zag diagonal PPset Trails that are the hallmark of the PTOP.

14. For every subsequent vertical PPset match, the Trail increases by one (1) PPset.

15. The rate of such PPset Trail growth far exceeds the PRIME Gap rate.

16. The zig-zag diagonal PPset Trails combine horizontally on the PTOP to give the ∑# of PPsets whose ∑s = The EVEN WIN.

17. More than simply proving the Goldbach Conjecture, the PTOP hidden within the BIM reveals a new NPS connection of the PRIMES: PRIMES + PRIMES = 90° R-angle isosceles triangles.

18. The entire BIM, including the ISL—Pythagorean Triples—and, PRIMES, is based on 90° R-triangles!

19. Similar to how the grid cell values of the Axis, PD and IG of the Pythagorean Triples reveal additional, intimate connections within the BIM, so too do the PPsets: the 1^st PRIME values of each set points to the # of STEPS from the PD that intersects the given EVEN WIN (Axis²), on a straight line path back to its Axis, at the apex where its other PPset member intersects — this is no mere coincidence — and that apex is, of course, the 90° R-angle isosceles triangle that results. For example: EVEN = 24, Axis = 12, PD = 144, # of STEPS from PD towards Axis = 5 for the 5+19 PPset and 7 for the 7+17 PPset, and 11 for the 11+13 PPset that each forms the EVEN 24.

20. As the bifurcation concatenation of the PS — 3,5,7,11,13,17,19,..— with the same base PS — 3,5,7,11,13,17,19,…— of those EVEN WIN — that when “3” is subtracted, the remainder is a next-in-the-sequence PRIME — remains one of a similar split with the Pythagorean Triples: for every “Primitive” parent PT, there are multiple “Non-Primitive” child PTs and it is the PPTs (Primitive Pythagorean Triples) that ultimately form the inter-connectedness of ALL PTs back to the original PPT — the 3-4-5. With the PRIMES, one raises the question: why are these “EVENS” preselected to be the “parent” EVENS forming the “beginning” or “start” of every PPset Trail with all other EVENS hitching on to that Trail further down the sequence?

21. Another set of STEPS (S) from the core Axis value , directly on the Axis, identifies each symmetrical pair of a given PPset that forms that given EVEN. These STEPS may by Universally calculated from the EVENS, P₁ and P₂ values. Examples are given further down.

As every new discovery unlocks many more questions, it follows that the details of the PTOP and BIM visualizations should both satisfy the proof offered and, more importantly, provide provocative data that will advance the field for the next researcher!

There are five sets of data to consider:

1. BIM (BBS-ISL Matrix): grid visualizations that overview the entire work

   • Fig. 1-PTOP: Periodic Table Of PRIMES (100, original)
   • Fig. 2-PTOP: Periodic Table Of PRIMES (100, annotated)
   • Fig. 3-PTOP: Periodic Table Of PRIMES (100, upgraded)
   • Fig. 4-PTOP: Periodic Table Of PRIMES (200, annotated)
   • Fig. 5-BIM: Symmetrical STEPS of the PPsets for EVEN 24 (original)
   • Fig. 6-BIM: Symmetrical STEPS of the PPsets for EVEN 24 (annotated)
   • Fig. 7-Table 46: Symmetrical STEPS of the PPsets for EVEN 128 annotated snapshot
   • Fig. 8-BIM: Symmetrical STEPS of the PPsets for EVEN 126 snapshot
   • Fig. 9-BIM: Symmetrical STEPS of the PPsets for EVEN 126
   • Fig. 10-BIM: Symmetrical STEPS of the PPsets for EVEN 128 snapshot
   • Fig. 11-BIM: Symmetrical STEPS of the PPsets for EVEN 128
   • Fig. 12- animated gif of video (below)
   • Video: PTOP rule 169-170 Annotated

2. PTOP: the actual Table

   ​
   • Table 34: Original PTOP from 2009 (EVENS 6-100)
   • Table 35: Upgraded PTOP (EVENS 6-200)
   • Table 36: Upgraded PTOP (EVENS 6-400)
   • Table 37: Upgraded PTOP (work in progress, EVENS 6-1000)

3. PTOP: Analysis

• Table 38: Distribution and NPS of the PPset Trails (EVENS 6-404)
• Table 39: Distribution and NPS of the PPset Trails (EVENS 6-914)
• Table 40: Distribution and NPS of the PPset Trails (EVENS 6-2360 and up and up)
• Table 41: Bifurcation Addition and PPset ∑s

4. Reference

   • Table 42: PRIME Gaps
   • Table 43: PRIME Partitions = PPsets per EVENS (4-2000)
   • Table 44: Summaries of Table 43 (EVENS 4-2000)
   • Table 45: Equations for PTOP Tables 38-41

5. PRIME PPset Trails

   ​
   • Table 46: PRIME PPset Trails (3-10007 and up)
   • Table 47: PRIME PPset Trails simplified and extended Table 46 (3-568201)
   • Table 48: Working example: EVEN 8872, core Axis 4436 with 93 PPsets.
   • Table 49: PRIME PPset Trails & EVENS divisible by 6,12 or 24 (3-10007 and up).

The DATA has been grouped in its own section down below. It is highly recommended that you, the reader, preview first — review, thereafter.

~~~~

PTOP: Periodic Table Of PRIMES & theGoldbach ConjectureINTRODUCTIONSTATEMENT: Layout & EssentialsThere are five sets of data to consider:FINDINGS & PROOFSUMMARY and CONCLUSIONDATA: Images and TablesDATA: Images and TablesREFERENCES

FINDINGS & PROOF

The definitive proof depends on demonstrating that the PPset Trails grow and extend to cover the “next” EVENS at a rate that exceeds the Prime Gap rate. PRIMES Sequence, PS is the key.

The PS is well established. How many primes are there?

As every beginning PPset — of 3,P₂ — forms from successively increasing the 2^nd PRIME by the next number in the PS (3,5,7,11,13,17,…), the Trail formed by bifurcating off from that point increases the total number sum (∑) of PPsets by one, i.e. Trail, Trail+1, (Trail+1)+1, ((Trail+1)+1)+1,…

A necessary, and sufficient, condition is that the Trail lengths — i.e. the total number of PPsets, in their overlapping aggregate, always exceed both the number of PRIMES and their Gaps for any and all numbers.

Specifically from Table 46 and Fig.__, locate a P₂ PRIME with a large Gap, e.i. P₂= 23, Gap=6.

line#———PRIMES, P₂—∑#set/Trail—PRIME Gap—∆Trail-Gap— EVEN:∑#sets—EC——E_ending——EVEN, E

We see that 23 has 8 PPsets in its Trail, it has a Gap of 6 to the next PRIME (for a ∆=2). It’s EVEN=26 and there are 3 PPsets (going horizontally across the PTOP) that will make 26. The 11 is the number of EVENS Covered (EC) as the PPsets zig-zag diagonally down the PTOP. If we add double this EC - 1 to the EVEN 26, we get the EVEN Ending of 46 as: 2(11-1) + 26 = 46. So this Trail alone inclusively covers the EVENS 26-46, although their will be some holes. To fill the holes, we look at the Trails that started up above this 23 Trail.

For the Specific set of equations exclusively for the 3,P₂ sets, where ∆ = P₂ - 3 (see Table 45: Equations):

2EC = P₂ -1

Ending EVEN Covered = E_e = 2(EC - 1) + EVEN

We can readily see that the 17 and 19 Trails, when their EVENS are added to their 2EC-1, respectively, will equal or exceed the EVEN of the NEXT PRIME up by the GAP=6.

Take Trail 17: EVEN = 20 with EC = 8. 2(8-1) + 20 = 34.

Take Trail 19: EVEN = 22 with EC = 9. 2(9-1) + 22 = 38.

And these Trails 17 and 19 will overlap the Trail 23 EC span, filling in any and all holes between it and the next Trail 29.

For completeness, we can see that Trail 13, while reaching Trail 23 at it’s start, does not overlap any further.

Take Trail 13: EVEN = 16 with EC = 6. 2(6-1) + 16 = 26. We also can see that 13 + 13 = 26.

~~~

The PPset Trails for EVEN 26 gives 3 PPsets via the overlapping Trails 17, 19 and 23, AND, their overlapping Trails extend past the next 3 EVENS (28, 30 and 32), ensuring that they are “covered” with PPsets in the Gap jump to the next PRIME (29). All very neat and clean.

• • The actual PPsets ∑s equal the EVEN 26:

  • • 3 + 23 = 26
    • 7 + 19 = 26
    • 13 + 13 = 26

  • which comes about more simply as the core Axis value ± steps away:

  • • 13 ± 10 = 3 and 23
    • 13 ± 6 = 7 and 19
    • 13 ± 0 = 13 and 13
    • where the core Axis value = 1/2 the EVEN, or 26/2 = 13 and we locate it on Column 2, PRIMES (line 5) for reference. It is not actually the PRIME, as we will see. Here it happens to land directly on the PRIME 13 as it is one of the PPsets. Other times it may well be somewhere between two PRIMES in this column. We will clarify this further on, but for now it is a reference to numbers on the Axis of the BIM that, being core Axis values, are key in showing that the PPsets will ALWAYS have their Pair members symmetrically located on either side of this reference marker.

But what about a larger Gap? How about Trail 113, EVEN = 116 and a Gap of 14 to the next Trail 127? (See image from Table 36.)

Trail 113 is overlapped by Trails 73, 79, 97, 103, and 109 that account for the 6 PPsets that will form the EVEN = 116 (the 3,113 PPset is included in the tally and you can follow it in Table 43). Again, from Table 46:

What about the Gap = 14 going to the Trail 127, EVEN = 130? Overlapping Trails 71, 83, 89, 101,107 and 113 combine with the 127 to account for the 7 PPsets that will form EVEN = 130.

Of course to account for all the EVENS between Trail 113 and 127 — EVENS 116 to 130, we have Trails 59 to and including 127 in various combinations to account for that. We can easily see this on Table 46, lines 16-30, where each has an E_e (118–254) that meets or exceeds those EVENS (116–130). Altogether, there are some 80 PPsets that account for these 15 EVENS 116-130. Easily done.

What if you want to know specifically who covers, say EVEN 128? Looking at the full table, line 18, we see that Trail 67 and above have E_e > 128. How can we determine that the 19+109, 31+97 and 61+67 PPsets are the ones we are looking for?

Without even knowing the values for the Column 6 (Table 46): EVEN, ∑# of PPsets, one can:

• look at an expanded PTOP

• look at an expanded BIM

• calculate using this trick:

• • on Table 46, find where EVEN 128 would be located between line 17 and 18 as 1/2 of 128 = 64

  • see that PRIMES 61 & 67 on either side = 128 when added together and both are PRIMES

  • knowing from the BIM that ALL PPsets are symmetrical about the center Axis core that points to the 90° R-angled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 64 slot like the 61 & 67

  • the next pair out would be 31 & 97, followed by 19 & 109

  • this is easily calculated as the core Axis value ± steps away:

  • • 64 ± 3 = 61 and 67
    • 64 ± 33 = 31 and 97
    • 64 ± 45 = 19 and 109
    • where the steps away are literally the number of steps along the Axis to either side of the core Axis value that goes to a PPset

  • their ∑s equal the EVEN 128:

  • • 61 + 67 = 128

    • 31 + 97 = 128

    • 19 + 109 = 128.

      ​

    • We can confirm this on the PTOP (Table 35), and, of course, directly on the BIM. Tables 43 and 44 can also confirm ALL EVENS up to 2000. One can also use an online calculator to get the results.

  • it can be quite helpful to follow these examples directly on the BIM. The visualizations of how the symmetrical PPsets contributed from the separate, individual PRIME Trails line up their 90° R-angled isosceles triangles over the central core with each of their respective apexes inline and pointing towards the EVEN that they are forming. On the Axis, one can easily see the symmetrical steps from the core to each PPset.

• 

• 

• (See the PDF versions for clear details.)

• What if you want to know specifically who covers, say EVEN 126? Looking at the full table, line 18, we see that Trail 67 and above have E_e > 126. How can we determine that the

• • • • 13+113

      • 17+109

      • 19+107

      • 23+103

      • 29+97

      • 37+89

      • 43+83

      • 47+79

      • 53+73

      • 59+67

        PPsets are the ones we are looking for?

• Without even knowing the values for the Column 6 (Table 46): EVEN, ∑# of PPsets, one can:

• look at an expanded PTOP

• look at an expanded BIM

• calculate using this trick:

• • on Table 46, find where EVEN 128 would be located between line 17 and 18 as 1/2 of 126 = 63

  • see that PRIMES 59 & 67 on either side = 126 when added together and both are PRIMES

  • knowing from the BIM that ALL PPsets are symmetrical about the center Axis core that points to the 90° R-angled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 63 slot like the 59 & 67

  • the next pair out would be 53 & 73, followed by 47 & 79.

  • this is easily calculated as the core Axis value ± steps away:

  • • 63 ± 4 = 59 and 67

    • 63 ± 10 = 53 and 73

    • 63 ± 16 = 47 and 79

    • 63 ± 20 = 43 and 83

    • 63 ± 26 = 37 and 89

    • 63 ± 34 = 29 and 97

    • 63 ± 40 = 23 and 103

    • 63 ± 44 = 19 and 107

    • 63 ± 46 = 17 and 109

    • 63 ± 50 = 13 and 113

      ​

  • their ∑s equal the EVEN 126:

  • • 59 + 67 = 126
    • 53 + 73 = 126
    • 47 + 79 = 126
    • 43 + 83 = 126
    • 37 + 89 = 126
    • 29 + 97 = 126
    • 23 + 103 = 126
    • 19 + 107 = 126
    • 17 + 109 = 126
    • 13 + 113 = 126

• The key is seeing that the PPsets are symmetrical about the core Axis value of the EVEN and applying these sequential, linear # of steps out from the core in identifying the PPsets that form that EVEN! EVERY PPset that forms an EVEN is part of a symmetrical pair of PPs, that together forms the EVEN. This is a critical and paramount finding in the behavior of the PRIMES and in the PROOF of the strong form of the Goldbach Conjecture.

• The ONLY EVENS (>4) with less than 2 PPsets are the EVENS 6, 8 and 12. EVEN 6 has only 1 PPset of 3,3 and EVEN 8 has only 1 PPset of 3,5. The EVEN 8 is the beginning of the 3 + 5 Trail, while the EVEN 6 is BOTH the start and end of the 3 + 3 Trail that begins the PTOP. The EVEN 12 is the one isolated example where the middle of a PRIME Trail — here the 3 + 7 Trail — is the only PPset to form EVEN 12, as the 3 + 5 Trail is — being the early start of Trails — not yet sufficient enough in length to cover the EVEN 12 slot. All 3 EVENS — 6,8,12 — are the exceptions precisely because they are the very beginning of the PTOP where the PRIME Trails are first being established.

• So we are claiming here — and giving as proof — that ALL EVENS ≥ 14 (and including EVEN 10), have a MINIMUM OF 2 PPsets. It is within the geometry of the number distribution of simple, natural, Whole Integer Numbers (WIN) that ALL EVEN WINs shall have their EVEN/2, Axis core value be symmetrically flanked on either side by ODD WINs, and, as we have shown here on the PTOP and the PTOP as hidden within the BIM, will have two (2) or more ODD PPs pairs within that symmetrical flanking that will, as PPsets, form said EVEN.

• The symmetry simply falls out of the inherent inter-connection between the number values of each cell within the BIM. Every 1^st Parallel EVEN has a straight line path back to its Axis and along that path the cell values must reflect the product of the two Axis values located symmetrically from the core Axis value outward, such that at their 90° R-angle, isosceles intersection on that path, one finds that product.

• The real wonder — and part of this amazing new finding — is that when such Axis numbers are found that are PRIME, they ALWAYS have a symmetrical ODD # counterpart on the other side of their common, core Axis number: and at the minimum, two or more candidates will have their ODDs be PRIMES, thus completing two or more PPsets!
• To remember, the Axis core number may itself be either EVEN or ODD as it is simply 1/2 of the EVEN that the path points to from its 1^st Parallel Diagonal location. Also, the above rule does NOT mean that ALL ODD #s found on the Axis are part of the PPsets that form the EVEN. Even though symmetrical, and forming the same products on the path, BOTH PPset MEMBERS FORMING THE PAIR MUST BE PRIMES, e.i. an Axis 9, 15, 21,.. are NOT PRIME and NOT part of a PPset forming the EVEN. The geometry defines the algebra = algebraic geometry.

• The growth rate of the staggered, zig-zagging diagonal Trails of the PS ensures that the number of overlapping Trails will always exceed the growth rate of the PRIME Gaps by a sizable margin.

Towards the bottom of Table 46, lines 1092 - 1107, there is a good run of PRIME Gaps in close proximity. Below is the table in plain form with the same column headings:

Without even knowing the values for the Column 6 (Table 46) which have been inserted here: EVEN, ∑# of PPsets, one can take EVEN 8872:

• look at an expanded PTOP

• look at an expanded BIM

• calculate using this trick: (see *below for a simple Universal calculation method as shown in Table 45: Equations.)

• • on Table 46, find where EVEN 8872 would be located as the core Axis value between lines 601 and 602 as 1/2 of 8872 = 4436

  • see that PRIMES 4423 & 4441 on either side = 8864 when added together and both are PRIMES we are close

  • knowing from the BIM that ALL PPsets are symmetrical about the center core that points to the 90° R-angled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 4436 slot like the 93 PPsets that equal EVEN 8872:

  • • 5+8867
    • 11+8861
    • 23+8849
    • 41+8831
    • 53+8819
    • 89+8783
    • 131+8741
    • 173+8699
    • 179+8693
    • 191+8681
    • 263+8609
    • 359+8513
    • 443+8429
    • 449+8423
    • 503+8369
    • 509+8363
    • 599+8273
    • 641+8231
    • 653+8219
    • 701+8171
    • 761+8111
    • 863+8009
    • 953+7919
    • 971+7901
    • 1019+7853
    • 1031+7841
    • 1049+7823
    • 1181+7691
    • 1223+7649
    • 1229+7643
    • 1283+7589
    • 1289+7583
    • 1373+7499
    • 1439+7433
    • 1523+7349
    • 1619+7253
    • 1721+7151
    • 1871+7001
    • 1889+6983
    • 1901+6971
    • 1913+6959
    • 1973+6899
    • 2003+6869
    • 2039+6833
    • 2069+6803
    • 2081+6791
    • 2111+6761
    • 2153+6719
    • 2213+6659
    • 2273+6599
    • 2309+6563
    • 2351+6521
    • 2381+6491
    • 2399+6473
    • 2423+6449
    • 2543+6329
    • 2549+6323
    • 2609+6263
    • 2699+6173
    • 2729+6143
    • 2741+6131
    • 2819+6053
    • 2843+6029
    • 2861+6011
    • 2969+5903
    • 3011+5861
    • 3023+5849
    • 3089+5783
    • 3203+5669
    • 3221+5651
    • 3299+5573
    • 3371+5501
    • 3389+5483
    • 3491+5381
    • 3539+5333
    • 3593+5279
    • 3701+5171
    • 3719+5153
    • 3821+5051
    • 3833+5039
    • 3851+5021
    • 3863+5009
    • 3929+4943
    • 4001+4871
    • 4073+4799
    • 4079+4793
    • 4139+4733
    • 4229+4643
    • 4289+4583
    • 4349+4523
    • 4391+4481
    • 4409+4463
    • 4421+4451

  • as before, these can be simplified as:

    • 4436 ± 5 = 4431 and 4441. Together, 4431 + 4441 = 8872. See Table 48 for the complete list.
  • • 4436 ± 11
    • 4436 ± 23
    • 4436 ± 41
    • 4436 ± 53
    • 4436 ± 89
    • 4436 ± 131
    • 4436 ± 173
    • 4436 ± 179
    • 4436 ± 191
    • 4436 ± 263
    • 4436 ± 359
    • 4436 ± 443
    • 4436 ± 449
    • 4436 ± 503
    • 4436 ± 509
    • 4436 ± 599
    • 4436 ± 641
    • 4436 ± 653
    • 4436 ± 701
    • 4436 ± 761
    • 4436 ± 863
    • 4436 ± 953
    • 4436 ± 1019
    • 4436 ± 1031
    • 4436 ± 1049
    • 4436 ± 1181
    • 4436 ± 1223
    • 4436 ± 1229
    • 4436 ± 1283
    • 4436 ± 1289
    • 4436 ± 1373
    • 4436 ± 1439
    • 4436 ± 1523
    • 4436 ± 1619
    • 4436 ± 1721
    • 4436 ± 1871
    • 4436 ± 1889
    • 4436 ± 1901
    • 4436 ± 1913
    • 4436 ± 1973
    • 4436 ± 2003
    • 4436 ± 2039
    • 4436 ± 2069
    • 4436 ± 2081
    • 4436 ± 2111
    • 4436 ± 2153
    • 4436 ± 2213
    • 4436 ± 2273
    • 4436 ± 2309
    • 4436 ± 2351
    • 4436 ± 2381
    • 4436 ± 2399
    • 4436 ± 2423
    • 4436 ± 2543
    • 4436 ± 2549
    • 4436 ± 2609
    • 4436 ± 2699
    • 4436 ± 2729
    • 4436 ± 2741
    • 4436 ± 2819
    • 4436 ± 2843
    • 4436 ± 2861
    • 4436 ± 2969
    • 4436 ± 3011
    • 4436 ± 3023
    • 4436 ± 3089
    • 4436 ± 3203
    • 4436 ± 3221
    • 4436 ± 3299
    • 4436 ± 3371
    • 4436 ± 3389
    • 4436 ± 3491
    • 4436 ± 3539
    • 4436 ± 3593
    • 4436 ± 3701
    • 4436 ± 3719
    • 4436 ± 3821
    • 4436 ± 3833
    • 4436 ± 3851
    • 4436 ± 3863
    • 4436 ± 3929
    • 4436 ± 4001
    • 4436 ± 4073
    • 4436 ± 4079
    • 4436 ± 4139
    • 4436 ± 4229
    • 4436 ± 4289
    • 4436 ± 4349
    • 4436 ± 4391
    • 4436 ± 4409
    • 4436 ± 4421

• • • ​

We can further simplify by applying these Universal (vs Specific, see Table 45:) equations:

• By definition, EVEN = P₁ + P₂ = PPset
• 

• let S = steps, E = EVEN = 2(core Axis value) = 2(A_x), as E/2 = A_x

• 

  ​

• S = P₂ - E/2 = P₂ - A_x

• 

• the PPset for a given EVEN:

• ​

• P₂ = S + E/2 = S + A_x

• 

  ​

• P₁ = P₂ - (2S)

• 

  ​

• 2S = P₂ - P₁

• 

  ​

• S = A_X - P₁

• 

  ​