## Preliminary Theory of physical Volume December 2020

Revised series at end of this paper: step:1300000928 Prime:29999882983 Sum:1.905670e+19 Lp:92.254487 idx:68 ### Introduction

What do three dimensional Prime Numbers [3DPN] and their Sums, actually mean in the 3D world?
Do they have a physical fundamental meaning?

This proof will show how the running sum of sequential PNs can fill/generate all volumes precisely.

Since the sum of these 3DPN increases indefinitly, all volumes are generated exactly as the next PN times a constant is accumulated.

The table's Top Plane [TP] column is an exact edge value of the dodecahedron this unique sum generates at that step.

With the TP in Lp form, the edge value is calculated by raising P to TP power (1.618...^TP).
The running sum shown in the revised series at the end of this proof forms a twelveth of a Dodecahedron, as shown on the right.

The dodecahedron volume is given by the formula of the edge length cubed times a derived constant hvc;
hvc = (( 7 * sqr(5) + 15 ) / 4) = 7.6631189606246~ whose Lp log is 4.23185585.

In Log form the formula is three times the TP plus the log of the above derived constant; so the Vol = P^(3*TP + 4.23185585) .
-------------------

### Theorem

For any volume of arbitrary size, there is a sequential 3DPN series sum, that times a geometrical constant
forms a precise dodecahedron shape that contains that volume within it. The following three examples are from table entries 28, 46 and 67 .
-----------------------------------------------------------------------------------------

#### 3DPN table

 IDX 3DPN Sum Vol = 3DPN * (P^5 + 1) Lp of Vol minus Lp(hvc) TP: Cube root of Vol/hvc Edge 28 1.20535677795e+11 1,457296,828826 58.20223629158205 53.97038044381005 17.99012681460363 5750.61312~ 46 6.96410239432319e+14 8419718145303998 76.20210235613656 71.97024650836457 23.99008216945485 103188.340~ 67 1.704528936160781980e+19 2.060804528405e20 97.20209662016157 92.97024077238957 30.99008023151986 2996013~

Therefore, entry 28 contains all dodecahedrons of edge less than 5750 units and 46 contains edges less than 103188 units.

Since there are infinite Prime Numbers,all Volumes are generated from this formula:
Note that the volume sum steps as each larger PN times P^5 + 1, is added to the running sum.

Therefore, every possible Volume is contained within the unending sequence of 3DPN sums times (P^5 + 1 ).

-------------------------------------------------------------------------------------

### Conclusions

One can now see why the RH built on knowing the EXACT number of PN's at every sequencial step, comes from the regularity of the 3DPN's, and their sums.

Additionally, the running sum is both absolute and relative making for any scale choice.
For example, the unit of volume can be set to any desired value P^0 = 1 ie. one cm, meter, foot, yard or one lightyear among others.

----------------------------------------------------------------------

### Table: The first 1300 Million Prime Sums

1ST 1300 Million Prime Sums arranged by Lp, the log of the golden ratio! December2020 -- twin PN count as twpn

0 Step:1 PN:2 Sum:2.000000000000000e+00 its Lp:1.440420 & TOP Plane:A0.000000000
1 Step:168 PN:997 Sum:7.612700000000000000e+04 its Lp:23.358026 & TOP Plane:A8.10185959234878 (Step*PN)/Sum=2.200218 tpncnt:14

2 Step:343 PN:2309 Sum:3.632880000000000000e+05 its Lp:26.605645 & TOP Plane:A9.18439942787884 (Step*PN)/Sum=2.180053 tpncnt:32
3 Step:474 PN:3361 Sum:7.342290000000000000e+05 its Lp:28.067839 & TOP Plane:A9.67179744006840 (Step*PN)/Sum=2.169778 tpncnt:42
4 Step:585 PN:4261 Sum:1.158040000000000000e+06 its Lp:29.014747 & TOP Plane:A9.98743337353519 (Step*PN)/Sum=2.152503 tpncnt:53

5 Step:741 PN:5641 Sum:1.931106000000000000e+06 its Lp:30.077406 & TOP Plane:A10.34165296961796 (Step*PN)/Sum=2.164553 tpncnt:64
6 Step:914 PN:7129 Sum:3.034407000000000000e+06 its Lp:31.016542 & TOP Plane:A10.65469814637067 (Step*PN)/Sum=2.147341 tpncnt:78
7 Step:1146 PN:9241 Sum:4.938359000000000000e+06 its Lp:32.028605 & TOP Plane:A10.99205274861029 (Step*PN)/Sum=2.144475 tpncnt:94

---- Note the three Lp step sequences have a plus one Top Plane sequence

8 Step:1432 PN:11941 Sum:7.962615000000000000e+06 its Lp:33.021358 & TOP Plane:A11.32297033960121 (Step*PN)/Sum=2.147474 tpncnt:113
9 Step:1795 PN:15361 Sum:1.291906600000000000e+07 its Lp:34.027042 & TOP Plane:A11.65819813743956 (Step*PN)/Sum=2.134287 tpncnt:130
10 Step:2261 PN:19993 Sum:2.115119400000000000e+07 its Lp:35.051522 & TOP Plane:A11.99969165263516 (Step*PN)/Sum=2.137192 tpncnt:164

11 Step:2841 PN:25801 Sum:3.441352200000000000e+07 its Lp:36.063038 & TOP Plane:A12.33686347967039 (Step*PN)/Sum=2.129995 tpncnt:200
12 Step:3531 PN:32941 Sum:5.468093800000000000e+07 its Lp:37.025328 & TOP Plane:A12.65762700029903 (Step*PN)/Sum=2.127152 tpncnt:245
13 Step:4435 PN:42409 Sum:8.874283999999999651e+07 its Lp:38.031595 & TOP Plane:A12.99304933599878 (Step*PN)/Sum=2.119426 tpncnt:306

14 Step:5619 PN:55333 Sum:1.466037379999999939e+08 its Lp:39.074775 & TOP Plane:A13.34077599322450 (Step*PN)/Sum=2.120793 tpncnt:378
15 Step:6975 PN:70381 Sum:2.316978559999999925e+08 its Lp:40.025917 & TOP Plane:A13.65782340501167 (Step*PN)/Sum=2.118740 tpncnt:461
16 Step:8750 PN:90373 Sum:3.742634550000000308e+08 its Lp:41.022414 & TOP Plane:A13.98998883371674 (Step*PN)/Sum=2.112853 tpncnt:567

17 Step:11007 PN:116533 Sum:6.075856100000000151e+08 its Lp:42.029316 & TOP Plane:A14.32562280840800 (Step*PN)/Sum=2.111108 tpncnt:702
18 Step:13809 PN:149521 Sum:9.801767119999999704e+08 its Lp:43.023139 & TOP Plane:A14.65689742371244 (Step*PN)/Sum=2.106493 tpncnt:858
19 Step:17381 PN:192613 Sum:1.591226048000000010e+09 its Lp:44.030029 & TOP Plane:A14.99252731920554 (Step*PN)/Sum=2.103916 tpncnt:1061

20 Step:21825 PN:247393 Sum:2.568569733999999880e+09 its Lp:45.025109 & TOP Plane:A15.32422074876522 (Step*PN)/Sum=2.102085 tpncnt:1297
21 Step:27453 PN:318181 Sum:4.159501893999999738e+09 its Lp:46.026843 & TOP Plane:A15.65813198752186 (Step*PN)/Sum=2.100017 tpncnt:1605
22 Step:34525 PN:408913 Sum:6.727916737999999896e+09 its Lp:47.026133 & TOP Plane:A15.99122869048541 (Step*PN)/Sum=2.098379 tpncnt:1961

23 Step:43400 PN:524413 Sum:1.086812765499999980e+10 its Lp:48.022719 & TOP Plane:A16.32342397007215 (Step*PN)/Sum=2.094153 tpncnt:2419
24 Step:54624 PN:674161 Sum:1.759190548699999927e+10 its Lp:49.023535 & TOP Plane:A16.65702926834385 (Step*PN)/Sum=2.093313 tpncnt:2971
25 Step:68739 PN:865261 Sum:2.845215661600000225e+10 its Lp:50.022648 & TOP Plane:A16.99006699771006 (Step*PN)/Sum=2.090428 tpncnt:3656

26 Step:86545 PN:1111213 Sum:4.604455455799999647e+10 its Lp:51.023009 & TOP Plane:A17.32352066514900 (Step*PN)/Sum=2.088628 tpncnt:4531
27 Step:108977 PN:1426753 Sum:7.449776016800000332e+10 its Lp:52.022901 & TOP Plane:A17.65681779124404 (Step*PN)/Sum=2.087086 tpncnt:5579
28 Step:137250 PN:1831129 Sum:1.205356777950000018e+11 its Lp:53.022828 & TOP Plane:A17.99012681460363 (Step*PN)/Sum=2.085046 tpncnt:6947   ******

29 Step:172903 PN:2350333 Sum:1.950372002079999819e+11 its Lp:54.022896 & TOP Plane:A18.32348279576891 (Step*PN)/Sum=2.083601 tpncnt:8604
30 Step:217882 PN:3015841 Sum:3.156775090870000049e+11 its Lp:55.023558 & TOP Plane:A18.65703711001857 (Step*PN)/Sum=2.081547 tpncnt:10556
31 Step:274550 PN:3868849 Sum:5.106837654689999670e+11 its Lp:56.023179 & TOP Plane:A18.99024407328614 (Step*PN)/Sum=2.079942 tpncnt:13094

32 Step:346018 PN:4961941 Sum:8.261694007050000437e+11 its Lp:57.022842 & TOP Plane:A19.32346482187382 (Step*PN)/Sum=2.078171 tpncnt:16163
33 Step:436193 PN:6364741 Sum:1.336711349365999922e+12 its Lp:58.022750 & TOP Plane:A19.65676767408759 (Step*PN)/Sum=2.076930 tpncnt:19984
34 Step:549972 PN:8162701 Sum:2.162873896773000160e+12 its Lp:59.022778 & TOP Plane:A19.99011045545392 (Step*PN)/Sum=2.075598 tpncnt:24773

35 Step:693568 PN:10467241 Sum:3.499987074869000207e+12 its Lp:60.023006 & TOP Plane:A20.32351971197980 (Step*PN)/Sum=2.074220 tpncnt:30666
36 Step:874687 PN:13418149 Sum:5.662214663291999750e+12 its Lp:61.022682 & TOP Plane:A20.65674498361379 (Step*PN)/Sum=2.072807 tpncnt:37944
37 Step:1103357 PN:17201581 Sum:9.162041406985999492e+12 its Lp:62.022770 & TOP Plane:A20.99010747310953 (Step*PN)/Sum=2.071534 tpncnt:47097

38 Step:1391984 PN:22045273 Sum:1.482426922476300097e+13 its Lp:63.022738 & TOP Plane:A21.32343028459634 (Step*PN)/Sum=2.070029 tpncnt:58379
39 Step:1756360 PN:28255741 Sum:2.398633007205899921e+13 its Lp:64.022752 & TOP Plane:A21.65676819834870 (Step*PN)/Sum=2.068981 tpncnt:72407
40 Step:2216395 PN:36211993 Sum:3.880989011430999963e+13 its Lp:65.022709 & TOP Plane:A21.99008712444184 (Step*PN)/Sum=2.068032 tpncnt:89914

41 Step:2797296 PN:46393513 Sum:6.279528026238100138e+13 its Lp:66.022694 & TOP Plane:A22.32341559264917 (Step*PN)/Sum=2.066658 tpncnt:111833
42 Step:3530932 PN:59442109 Sum:1.016066226607989985e+14 its Lp:67.022729 & TOP Plane:A22.65676068520963 (Step*PN)/Sum=2.065673 tpncnt:138946
43 Step:4457439 PN:76142389 Sum:1.643998489623140078e+14 its Lp:68.022690 & TOP Plane:A22.99008087267042 (Step*PN)/Sum=2.064479 tpncnt:172855

44 Step:5627762 PN:97537021 Sum:2.660051900037529878e+14 its Lp:69.022695 & TOP Plane:A23.32341588989360 (Step*PN)/Sum=2.063550 tpncnt:214958
45 Step:7106178 PN:124926301 Sum:4.304053507846309803e+14 its Lp:70.022694 & TOP Plane:A23.65674908188062 (Step*PN)/Sum=2.062587 tpncnt:267879
46 Step:8973948 PN:159990121 Sum:6.964102394323189743e+14 its Lp:71.022694 & TOP Plane:A23.99008216945485 (Step*PN)/Sum=2.061634 tpncnt:333622   ******

47 Step:11333862 PN:204873961 Sum:1.126813352672437020e+15 its Lp:72.022690 & TOP Plane:A24.32341422115625 (Step*PN)/Sum=2.060690 tpncnt:415263
48 Step:14315843 PN:262319713 Sum:1.823222641810188070e+15 its Lp:73.022690 & TOP Plane:A24.65674768298508 (Step*PN)/Sum=2.059720 tpncnt:517033
49 Step:18084258 PN:335866801 Sum:2.950043737855964899e+15 its Lp:74.022695 & TOP Plane:A24.99008278545054 (Step*PN)/Sum=2.058919 tpncnt:644678

50 Step:22846807 PN:429975349 Sum:4.773256767084880173e+15 its Lp:75.022689 & TOP Plane:A25.32341404805843 (Step*PN)/Sum=2.058042 tpncnt:803458
51 Step:28866398 PN:550421233 Sum:7.723301611883856356e+15 its Lp:76.022692 & TOP Plane:A25.65674827161992 (Step*PN)/Sum=2.057239 tpncnt:1001863
52 Step:36475382 PN:704527261 Sum:1.249655687612378824e+16 its Lp:77.022691 & TOP Plane:A25.99008118161241 (Step*PN)/Sum=2.056399 tpncnt:1250100

53 Step:46094177 PN:901747333 Sum:2.021986935966312740e+16 its Lp:78.022692 & TOP Plane:A26.32341504909067 (Step*PN)/Sum=2.055666 tpncnt:1559971
54 Step:58254573 PN:1154047033 Sum:3.271636217210404357e+16 its Lp:79.022688 & TOP Plane:A26.65674682199527 (Step*PN)/Sum=2.054890 tpncnt:1946687
55 Step:73629386 PN:1476844081 Sum:5.293625230359453417e+16 its Lp:80.022690 & TOP Plane:A26.99008102316890 (Step*PN)/Sum=2.054152 tpncnt:2430290

56 Step:93069731 PN:1889788921 Sum:8.565255162417989050e+16 its Lp:81.022688 & TOP Plane:A27.32341351671961 (Step*PN)/Sum=2.053437 tpncnt:3034819
57 Step:117652438 PN:2418028969 Sum:1.385888708599541860e+17 its Lp:82.022690 & TOP Plane:A27.65674750536235 (Step*PN)/Sum=2.052741 tpncnt:3789741
58 Step:148739891 PN:3093685981 Sum:2.242413148542554700e+17 its Lp:83.022688 & TOP Plane:A27.99008025591439 (Step*PN)/Sum=2.052051 tpncnt:4735049

59 Step:188056140 PN:3957869173 Sum:3.628302638901245082e+17 its Lp:84.022689 & TOP Plane:A28.32341396109961 (Step*PN)/Sum=2.051377 tpncnt:5914645
60 Step:237782461 PN:5063171881 Sum:5.870714289213775191e+17 its Lp:85.022688 & TOP Plane:A28.65674697561949 (Step*PN)/Sum=2.050744 tpncnt:7392255
61 Step:300679658 PN:6476688949 Sum:9.499018215457072947e+17 its Lp:86.022689 & TOP Plane:A28.99008052460542 (Step*PN)/Sum=2.050116 tpncnt:9240610

62 Step:380241012 PN:8284240273 Sum:1.536972687387462705e+18 its Lp:87.022688 & TOP Plane:A29.32341352179344 (Step*PN)/Sum=2.049489 tpncnt:11552363
63 Step:480888455 PN:10595706001 Sum:2.486874021830610000e+18 its Lp:88.022688 & TOP Plane:A29.65674684784500 (Step*PN)/Sum=2.048899 tpncnt:14445080
64 Step:608217669 PN:13551217081 Sum:4.023846571664937306e+18 its Lp:89.022688 & TOP Plane:A29.99008016028027 (Step*PN)/Sum=2.048311 tpncnt:18064613

65 Step:769312273 PN:17330083753 Sum:6.510722003020669334e+18 its Lp:90.022688 & TOP Plane:A30.32341365155966 (Step*PN)/Sum=2.047737 tpncnt:22597343
66 Step:973137440 PN:22161454789 Sum:1.053456735516101681e+19 its Lp:91.022688 & TOP Plane:A30.65674684437370 (Step*PN)/Sum=2.047179 tpncnt:28273103
67 Step:1231043265 PN:28338247669 Sum:1.704528936160781980e+19 its Lp:92.022688 & TOP Plane:A30.99008023151986 (Step*PN)/Sum=2.046642 tpncnt:35380403  ******

-end- step:1300000928 Prime:29999882983 Sum:1.905670e+19 Lp:92.254487 idx:68 Base=27.000000 bias=0.000000 Fiby:1 skips 0

SUM of all PN's is 19056695307656212105.

---RUN Check Values
Step: 78498 PN : 999983 Sum: 3.755040E+10 Lp: 50.599234
Step: 100000 PN : 1299709 Sum: 6.226070E+10 Lp: 51.650011
Step: 50847534 PN : 999999937 Sum: 2.473951E+16 Lp: 78.441917
Step: 455052511 PN : 9999999967 Sum: 2.220822E+18 Lp: 87.787554

author: RD O'Meara Oak Park, IL.

Email of Author: 'RDo.meara@mister-computer.net'

JID's SLOPE: The Universal Slope of Volume, both Mathematically and Physically   AKA, the Rydberg constant of 1.0973~!

Short proof of Fermat's FLT: A proof based on power sequences infinite transcendental Logs.

# DATA SANITY

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