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From: rom126 on 24 Feb 2010 11:26
A pattern found in first 25 million Prime Number sums -
using the golden ratio log Lp!


----------Assume Prime Numbers are Three Dimensional

ABST: Prime Number sums reveal a 3D pattern - using the golden ratio log!

----------Introduction

It is well known that Prime Number attributes are logarithmic and that
they form an upward spiral in some fashion.

This 3D pattern was found by assuming Primes are cubic numbers
and that their sum represents the interior volume of the Prime spiral.

For example, the sum of the first 25 primes is assumed to be 160 cubic units,
and first 168 sums to 76,127 cubic units, etc.

The Golden ratio was chosen because its has all three ratios simultaneously:
length increases by P^1=1.618... , area by P^2=2.618... and volume by
P^3=4.23606... ;
also note that P^2 = P^1 + 1 and P^3 = 4 + P^-3.

Lp is the log to base ( sqrt(5) + 1) / 2 ) = 1.618.... Lp(X) = log( X )
/ log(1.618....) .

Here are some common Lp examples/values:
Lp(123) = 10.000137.. ; and the inverse Px(10) = 122.991....
Lp(199) = 10.9999..., Lp(pi) = 2.37885~, Lp(e) = 2.078087~, Lp(2) =
1.44042~.

Lucas numbers represent the integer steps of Px(n) series; 123 is the 10th.

-----------

Results for the first 25Million Primes Sums

A regular pattern was found by summing the first 25KK Primes and noting sums
based on Lp unit increments.
That perspective reveals a regular three step pattern documented in the table
below as a sequence that approaches two.

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

The six column table lists the step count, the Prime at that step,
the ratio to Prime back 3, the sum of all Primes to that step, its log using
the golden ratio- Lp,
and the plus 3 Lp volume step ratio which approaches two from both above and
below.

The reason the Lp's can not increase by exactly one, is that Lp(primesum)
steps from less than one to over one,
based on the last prime integer added.

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

For example, the last two P^3 steps are
step:8975619 Prime:1.600215E8 Nratio:2.1012 Sum:6.9667E14 Lp:71.023491
P^3-step:2.013
step:18087624 Prime:3.359332E8 Nratio:2.0993 Sum:2.9511E15 Lp:74.023492
P^3-step:2.015

Where P^3 step is 18087624 / 8975619 = 2.015.
and the Nratio of the last Prime is 3.359332e8 / 1.600215e8 = 2.0993 .


-----------Conclusions

The Lp three step Pattern of the Prime Sums is clearly real and definitive;
its exact shape/structure remains a mystery for now.

This spiral pattern will obviously go on indefinitely,
so the only question left concerns the P^3 step and Nratio meeting or
crossing, as the 3D Prime Sums increase.

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

I credit my Computer System Analyses Skills and Tools, along with my
Geometrical insights, in making this discovery possible.


I am working on a "Geometry of the Prime Numbers" paper based on these
insights.
It uses the 12 sided Dodecahedron as the 3D model that has six axes at 60 deg
offsets.
It is tentatively titled "The Prime Numbers as Rain drops".

See my site for the latest material.
<A HREF="http://mister-computer.net/index.htm">WEB Site"</A>

RD OMeara Oak Park IL 24Feb2010

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

----------- Table of Prime sums at Golden Ratio Logs ---------

1ST 25Million Prime Sums arranged by Lp, the log of the golden ratio!
24Feb2010 - All Rights Reserved - RD OMeara - 1.primes3d(a)mister-computer.net


step:25 Prime:9.700000E1 Nratio:0.0000 Sum:1.0600E3 Lp:14.476004
P^3-step:1.000
step:168 Prime:9.970000E2 Nratio:0.0000 Sum:7.6127E4 Lp:23.358026
P^3-step:1.000
step:375 Prime:2.557000E3 Nratio:0.0000 Sum:4.4110E5 Lp:27.008980
P^3-step:1.000

step:468 Prime:3.323000E3 Nratio:0.0000 Sum:7.1415E5 Lp:28.010245
P^3-step:1.000
step:585 Prime:4.261000E3 Nratio:0.0000 Sum:1.1580E6 Lp:29.014747
P^3-step:1.000

step:731 Prime:5.527000E3 Nratio:2.1615 Sum:1.8752E6 Lp:30.016399
P^3-step:1.949
step:914 Prime:7.129000E3 Nratio:2.1454 Sum:3.0344E6 Lp:31.016542
P^3-step:1.953
step:1143 Prime:9.221000E3 Nratio:2.1640 Sum:4.9106E6 Lp:32.016913
P^3-step:1.954

step:1431 Prime:1.193900E4 Nratio:2.1601 Sum:7.9506E6 Lp:33.018239
P^3-step:1.958
step:1792 Prime:1.533100E4 Nratio:2.1505 Sum:1.2873E7 Lp:34.019618
P^3-step:1.961
step:2245 Prime:1.984300E4 Nratio:2.1519 Sum:2.0832E7 Lp:35.019955
P^3-step:1.964

step:2814 Prime:2.556100E4 Nratio:2.1410 Sum:3.3720E7 Lp:36.020740
P^3-step:1.966
step:3528 Prime:3.291700E4 Nratio:2.1471 Sum:5.4582E7 Lp:37.021569
P^3-step:1.969
step:4425 Prime:4.233100E4 Nratio:2.1333 Sum:8.8319E7 Lp:38.021647
P^3-step:1.971

step:5552 Prime:5.455900E4 Nratio:2.1345 Sum:1.4292E8 Lp:39.021933
P^3-step:1.973
step:6969 Prime:7.031300E4 Nratio:2.1361 Sum:2.3127E8 Lp:40.022128
P^3-step:1.975
step:8750 Prime:9.037300E4 Nratio:2.1349 Sum:3.7426E8 Lp:41.022414
P^3-step:1.977

step:10990 Prime:1.163410E5 Nratio:2.1324 Sum:6.0560E8 Lp:42.022534
P^3-step:1.979
step:13808 Prime:1.495190E5 Nratio:2.1265 Sum:9.8002E8 Lp:43.022822
P^3-step:1.981
step:17353 Prime:1.923070E5 Nratio:2.1279 Sum:1.5858E9 Lp:44.022979
P^3-step:1.983

step:21815 Prime:2.472790E5 Nratio:2.1255 Sum:2.5661E9 Lp:45.023107
P^3-step:1.985
step:27430 Prime:3.178570E5 Nratio:2.1259 Sum:4.1521E9 Lp:46.023185
P^3-step:1.987
step:34502 Prime:4.086310E5 Nratio:2.1249 Sum:6.7185E9 Lp:47.023227
P^3-step:1.988

step:43406 Prime:5.245190E5 Nratio:2.1212 Sum:1.0871E10 Lp:48.023321
P^3-step:1.990
step:54622 Prime:6.741310E5 Nratio:2.1209 Sum:1.7590E10 Lp:49.023376
P^3-step:1.991
step:68751 Prime:8.654090E5 Nratio:2.1178 Sum:2.8462E10 Lp:50.023406
P^3-step:1.993

step:86553 Prime:1.111339E6 Nratio:2.1188 Sum:4.6053E10 Lp:51.023410
P^3-step:1.994
step:108990 Prime:1.426933E6 Nratio:2.1167 Sum:7.4516E10 Lp:52.023418
P^3-step:1.995
step:137269 Prime:1.831399E6 Nratio:2.1162 Sum:1.2057E11 Lp:53.023427
P^3-step:1.997

step:172925 Prime:2.350631E6 Nratio:2.1151 Sum:1.9508E11 Lp:54.023446
P^3-step:1.998
step:217877 Prime:3.015791E6 Nratio:2.1135 Sum:3.1566E11 Lp:55.023459
P^3-step:1.999
step:274568 Prime:3.869111E6 Nratio:2.1127 Sum:5.1075E11 Lp:56.023463
P^3-step:2.000

step:346068 Prime:4.962677E6 Nratio:2.1112 Sum:8.2641E11 Lp:57.023466
P^3-step:2.001
step:436266 Prime:6.365773E6 Nratio:2.1108 Sum:1.3371E12 Lp:58.023472
P^3-step:2.002
step:550061 Prime:8.164027E6 Nratio:2.1101 Sum:2.1636E12 Lp:59.023476
P^3-step:2.003

step:693644 Prime:1.046846E7 Nratio:2.1094 Sum:3.5007E12 Lp:60.023479
P^3-step:2.004
step:874849 Prime:1.342099E7 Nratio:2.1083 Sum:5.6643E12 Lp:61.023480
P^3-step:2.005
step:1103540 Prime:1.720452E7 Nratio:2.1074 Sum:9.1651E12 Lp:62.023483
P^3-step:2.006

step:1392226 Prime:2.204914E7 Nratio:2.1062 Sum:1.4829E13 Lp:63.023486
P^3-step:2.007
step:1756660 Prime:2.826113E7 Nratio:2.1057 Sum:2.3994E13 Lp:64.023486
P^3-step:2.008
step:2216797 Prime:3.621858E7 Nratio:2.1052 Sum:3.8824E13 Lp:65.023488
P^3-step:2.009

------------------------------- inner two steps ommited for last three
step:4458271 Prime:7.615752E7 Nratio:2.1027 Sum:1.6446E14 Lp:68.023490
P^3-step:2.011

step:8975619 Prime:1.600215E8 Nratio:2.1012 Sum:6.9667E14 Lp:71.023491
P^3-step:2.013

step:18087624 Prime:3.359332E8 Nratio:2.0993 Sum:2.9511E15 Lp:74.023492
P^3-step:2.015

-end- step:24999000 Prime:4.728616E8 Sum:5.744761E15 Lp:75.407675 idx:51
bias=26.0000

------- end table --------

--
Regards from RD

--
Yours truly RD email mr.computer(a)pobox.com

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