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From: rom126 on 22 Apr 2010 15:46
ABSTRACT: Prime Number Sums reveal both a 2D and 3D pattern when stepped by
the golden ratio logarithm!


<CENTER><H3>Assumptions & Hypothesis</H3></CENTER>

It is well known that Prime Number [PN] attributes are logarithmic and that
they form an upward spiral in some fashion; probably the logarithmic one.

In re-examining the concepts of NUMBER, quantity is its understood
fundamental.
Therefore, assuming Prime Numbers are three dimensional in nature seems
conservative;
since Volume is the essence of 3D-NUMBER sums.

Prime Number Sums then take on special aspects with the assumption that Prime
Numbers are 3D in nature.
The closest analogy is that Prime Numbers are discrete liquid drops whose
values represent unique quantities of liquid as volume. Again, a 3D aspect
fundamental.

If the liquid analogy is correct, then PN sums should reveal their 3D nature
in some numerical fashion.
as 2D or area/surface values were proven shapeless outside of the Riemann Hyp
domain; so far.
After examining the first 100 million PN sums using all the common log bases,
no patterns emerge that support a 3D basis.
However, when the golden ratio is used as the log base [Lp], a clear and
repeating pattern in both 2D and 3D emerges; as graphed in the following
images.


<CENTER><H3>Results for the first 1300 Million Primes Sums</H3></CENTER>

The appended Table reveals a regular pattern.
It was found by summing the first 1300 Million Primes
and noting their sums based on Lp unit increments.

This P^1 or Lp +1 snapshot perspective reveals
two more resonances at P^3 and P^4 steps
that seem to slowly change predictably.




The six column Table lists the step count,
the Prime at that step: PN, the PN depth at that step,
the sum of all Primes to that step and its Golden Ratio log Lp,

and the TOP plane at that step derived from the growth of the last 3 sums.


For example, the Table's last five Lp+3 values are:

Step:73629340 PN:1476842981 PN Depth:3.227960e+08 Sum:5.293618e+16 its
Lp:80.0227 & TOP Plane:A27.004978 *-12
Step:148739879 PN:3093685613 PN Depth:6.756579e+08 Sum:2.242413e+17 its
Lp:83.0227 & TOP Plane:A28.004978 *-9
Step:300679579 PN:6476687027 PN Depth:1.413516e+09 Sum:9.499013e+17 its
Lp:86.0227 & TOP Plane:A29.004978 *-6
Step:608217668 PN:13551217079 PN Depth:2.955511e+09 Sum:4.023847e+18 its
Lp:89.0227 & TOP Plane:A30.004978 *-3
Step:1231043204 PN:28338246133 PN Depth:6.176792e+09 Sum:1.704529e+19 its
Lp:92.0227 & TOP Plane:A31.004978 *

For the last step, the Sum is 1.704529e+19 its Lp:92.0227 & TOP
Plane:A31.004978
And the PN Depth to A30.67's PN, is 28338246133 - 22161454589 =
6.176792e+09

The last 4 step sums are mapped between 5 top planes
at scales A27, A28, A29, A30 and A31 ; A27 is P^-4 smaller than A29.
Graphing A31down to A27 had 3 interior zones in a d2 array.

The last 3 step sums are mapped between 4 top planes
at scales A28, A29, A30 and A31 ; A31 is P^2 larger than A30.
Graphing A31 down to A28 had 3 interior zones in a d3 array.

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.



<IMG SRC="http://mister-computer.net/pixs/3Da1242.JPG"ALIGN=left BORDER=0
WIDTH=38% HEIGHT=62%>
<IMG SRC="http://mister-computer.net/pixs/2Da1244.JPG"ALIGN=right BORDER=0
WIDTH=38% HEIGHT=62%>

The 3D figure on the left illustrates the three zones between the four Top
Planes.
While the 2D figure on the right shows a linear four step pattern between the
last five Top Planes.




<CENTER><H3>Conclusions</H3></CENTER>

Both the three and four step Lp Patterns of the Prime Sums is clearly real and
definitive!
These patterns will obviously go on indefinitely,

Therefore Prime Numbers and their distribution as 3D sums is self-evident, as
witnessed by these Graphic Proofs!
It is TRUELY amazing that Prime Number Sums are simultaneously one, two and
three dimensional in nature!
But only if you use the Golden ratio as your Log base does the pattern emerge.



<CENTER><H3>Discussion</H3></CENTER>
I have a whole new respect for "complexity and the nature of 3D " after my
discovery.

My initial goal was to find the short form of FLT and maybe a backdoor into
the RH via the 1/2 axis .
It was possible that the RH's 1/2 critical line was just another form of "TWO
as a unique power" that was recently proven.
These data patterns might be used to prove the Riemann Hyp.; since they
allowed for many unique perspectives
through simultaneous 3D, 2D and 1D filters.



Other gems found in these patterns could be the basis of many kinds of PRNG's
and Encoders, simply by biasing the generator.


A possible Public Key system could be built on the subset of Triples: 3
unique values of Step;PN;SUM; when SUM is also a Prime!

The user would select several of these Triples from the large public list [ or
the first part could be locally re-generated in r/t ;
ie. 1127th Triple is PN:555491 its SUM:12.131607731e9 at Step:45700].
An algorithm would be used to input a key and generate a unique local
temporary master table file based on the Triples used.



<CENTER><H3>Golden Ratio Logs</H3></CENTER>

The 3D side view of the last nine steps is illustrated on the right.
<IMG SRC="http://mister-computer.net/pixs/3side1241.JPG"ALIGN=right BORDER=0
WIDTH=38% HEIGHT=62%>

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 also that it is close to 'e' ,
and P^3 = 4 + P^-3 or 4 = P^3 - 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(10) = 4.784972~ Lp(pi) = 2.37885~, Lp(e) = 2.078087~, Lp(2) = 1.44042~.
Px(7) = 29.03442~ , Lp(c) = 50.1313~ with P^0 = one centi-metre [cm]


Lucas numbers represent the integer steps of Px(n) series;
29, 47, 76, 123, 199 are the 7th, to 11th Lucas numbers.


The 3D rotated and 2D aligned pair are shown on the right.
<IMG SRC="http://mister-computer.net/pixs/3Ddual1245.JPG"ALIGN=right BORDER=0
WIDTH=30% HEIGHT=92%>

<CENTER><H3>Some spatial Perspectives</H3></CENTER>

I have assigned an arbitrary scale and dimension to the powers of Px,
a one centimeter reference was selected for our human scale.
<BR
More likely, it was probably a fm scale or lower for nuclear geometry .


P^0 = 1 cm & P^12 is 322~ cm, this makes P^31 equal to 3.010349e6 cm
and its area was A31 at 15.5913122 e12 cm^2 .



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


I am working on a "Geometry of the Prime Numbers" , based on these insights.

See my site for the latest material. <A
HREF="http://mister-computer.net/index.htm">WEB Site"</A>
RD OMeara Oak Park IL 15April2010

<HR
---------------------------TABLE-------------------------------------

1ST 1300 Million Prime Sums arranged by Lp, the log of the golden ratio!
14April2010 Author: RD_OMeara @ 60302 1.PrimeSums(a)Mister-Computer.Net

Step:25 PN:97 PN Depth:0.000000e+00 Sum:1.060000e+03 its Lp:14.476 & TOP
Plane:A0.000000
Step:168 PN:997 PN Depth:0.000000e+00 Sum:7.612700e+04 its Lp:23.358 & TOP
Plane:A0.000000
Step:343 PN:2309 PN Depth:0.000000e+00 Sum:3.632880e+05 its Lp:26.6056 & TOP
Plane:A0.000000
Step:467 PN:3319 PN Depth:0.000000e+00 Sum:7.108360e+05 its Lp:28.0006 & TOP
Plane:A0.000000
Step:584 PN:4259 PN Depth:9.400000e+02 Sum:1.153779e+06 its Lp:29.0071 & TOP
Plane:A10.139022

Step:730 PN:5521 PN Depth:1.262000e+03 Sum:1.869711e+06 its Lp:30.0103 & TOP
Plane:A10.371043
Step:913 PN:7127 PN Depth:1.606000e+03 Sum:3.027278e+06 its Lp:31.0117 & TOP
Plane:A10.669106
Step:1142 PN:9209 PN Depth:2.082000e+03 Sum:4.901431e+06 its Lp:32.013 & TOP
Plane:A11.002360

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

Step:1430 PN:11933 PN Depth:2.724000e+03 Sum:7.938735e+06 its Lp:33.0151 &
TOP Plane:A11.336286
Step:1791 PN:15329 PN Depth:3.396000e+03 Sum:1.285767e+07 its Lp:34.0171 &
TOP Plane:A11.670360
Step:2244 PN:19841 PN Depth:4.512000e+03 Sum:2.081249e+07 its Lp:35.018 &
TOP Plane:A12.003918

Step:2813 PN:25541 PN Depth:5.700000e+03 Sum:3.369459e+07 its Lp:36.0192 &
TOP Plane:A12.337553
Step:3527 PN:32911 PN Depth:7.370000e+03 Sum:5.454921e+07 its Lp:37.0203 &
TOP Plane:A12.671180
Step:4424 PN:42323 PN Depth:9.412000e+03 Sum:8.827671e+07 its Lp:38.0207 &
TOP Plane:A13.004574

Step:5551 PN:54547 PN Depth:1.222400e+04 Sum:1.428683e+08 its Lp:39.0211 &
TOP Plane:A13.337998
Step:6968 PN:70309 PN Depth:1.576200e+04 Sum:2.312054e+08 its Lp:40.0215 &
TOP Plane:A13.671369
Step:8749 PN:90371 PN Depth:2.006200e+04 Sum:3.741731e+08 its Lp:41.0219 &
TOP Plane:A14.004849

Step:10989 PN:116329 PN Depth:2.595800e+04 Sum:6.054898e+08 its Lp:42.0221 &
TOP Plane:A14.338229
Step:13806 PN:149497 PN Depth:3.316800e+04 Sum:9.797282e+08 its Lp:43.0222 &
TOP Plane:A14.671549
Step:17350 PN:192259 PN Depth:4.276200e+04 Sum:1.585260e+09 its Lp:44.0222 &
TOP Plane:A15.004855

Step:21811 PN:247229 PN Depth:5.497000e+04 Sum:2.565107e+09 its Lp:45.0223 &
TOP Plane:A15.338201
Step:27425 PN:317789 PN Depth:7.056000e+04 Sum:4.150598e+09 its Lp:46.0224 &
TOP Plane:A15.671566
Step:34496 PN:408533 PN Depth:9.074400e+04 Sum:6.716064e+09 its Lp:47.0225 &
TOP Plane:A16.004930

Step:43398 PN:524389 PN Depth:1.158560e+05 Sum:1.086708e+10 its Lp:48.0225 &
TOP Plane:A16.338276
Step:54612 PN:673979 PN Depth:1.495900e+05 Sum:1.758382e+10 its Lp:49.0226 &
TOP Plane:A16.671628
Step:68738 PN:865259 PN Depth:1.912800e+05 Sum:2.845129e+10 its Lp:50.0226 &
TOP Plane:A17.004955

Step:86537 PN:1111091 PN Depth:2.458320e+05 Sum:4.603567e+10 its Lp:51.0226
& TOP Plane:A17.338294
Step:108970 PN:1426693 PN Depth:3.156020e+05 Sum:7.448777e+10 its Lp:52.0226
& TOP Plane:A17.671627
Step:137244 PN:1831033 PN Depth:4.043400e+05 Sum:1.205247e+11 its Lp:53.0226
& TOP Plane:A18.004967

Step:172893 PN:2350237 PN Depth:5.192040e+05 Sum:1.950137e+11 its Lp:54.0226
& TOP Plane:A18.338301
Step:217836 PN:3015113 PN Depth:6.648760e+05 Sum:3.155388e+11 its Lp:55.0226
& TOP Plane:A18.671632
Step:274517 PN:3868279 PN Depth:8.531660e+05 Sum:5.105561e+11 its Lp:56.0227
& TOP Plane:A19.004971

Step:346004 PN:4961783 PN Depth:1.093504e+06 Sum:8.260999e+11 its Lp:57.0227
& TOP Plane:A19.338306
Step:436185 PN:6364637 PN Depth:1.402854e+06 Sum:1.336660e+12 its Lp:58.0227
& TOP Plane:A19.671641
Step:549959 PN:8162513 PN Depth:1.797876e+06 Sum:2.162768e+12 its Lp:59.0227
& TOP Plane:A20.004976

Step:693515 PN:10466471 PN Depth:2.303958e+06 Sum:3.499432e+12 its
Lp:60.0227 & TOP Plane:A20.338308
Step:874686 PN:13418147 PN Depth:2.951676e+06 Sum:5.662201e+12 its
Lp:61.0227 & TOP Plane:A20.671641
Step:1103334 PN:17201237 PN Depth:3.783090e+06 Sum:9.161646e+12 its
Lp:62.0227 & TOP Plane:A21.004975

Step:1391966 PN:22044949 PN Depth:4.843712e+06 Sum:1.482387e+13 its
Lp:63.0227 & TOP Plane:A21.338310
Step:1756332 PN:28255313 PN Depth:6.210364e+06 Sum:2.398554e+13 its
Lp:64.0227 & TOP Plane:A21.671643
Step:2216382 PN:36211757 PN Depth:7.956444e+06 Sum:3.880942e+13 its
Lp:65.0227 & TOP Plane:A22.004976

Step:2797290 PN:46393433 PN Depth:1.018168e+07 Sum:6.279500e+13 its
Lp:66.0227 & TOP Plane:A22.338310
Step:3530896 PN:59441509 PN Depth:1.304808e+07 Sum:1.016045e+14 its
Lp:67.0227 & TOP Plane:A22.671644
Step:4457435 PN:76142323 PN Depth:1.670081e+07 Sum:1.643995e+14 its
Lp:68.0227 & TOP Plane:A23.004977

Step:5627751 PN:97536821 PN Depth:2.139450e+07 Sum:2.660041e+14 its
Lp:69.0227 & TOP Plane:A23.338311
Step:7106165 PN:124926157 PN Depth:2.738934e+07 Sum:4.304037e+14 its
Lp:70.0227 & TOP Plane:A23.671644
Step:8973934 PN:159989891 PN Depth:3.506373e+07 Sum:6.964080e+14 its
Lp:71.0227 & TOP Plane:A24.004977

Step:11333855 PN:204873829 PN Depth:4.488394e+07 Sum:1.126812e+15 its
Lp:72.0227 & TOP Plane:A24.338311
Step:14315833 PN:262319557 PN Depth:5.744573e+07 Sum:1.823220e+15 its
Lp:73.0227 & TOP Plane:A24.671644
Step:18084223 PN:335866043 PN Depth:7.354649e+07 Sum:2.950032e+15 its
Lp:74.0227 & TOP Plane:A25.004977

Step:22846796 PN:429975173 PN Depth:9.410913e+07 Sum:4.773252e+15 its
Lp:75.0227 & TOP Plane:A25.338311
Step:28866367 PN:550420693 PN Depth:1.204455e+08 Sum:7.723285e+15 its
Lp:76.0227 & TOP Plane:A25.671644
Step:36475354 PN:704526727 PN Depth:1.541060e+08 Sum:1.249654e+16 its
Lp:77.0227 & TOP Plane:A26.004978

Step:46094125 PN:901746253 PN Depth:1.972195e+08 Sum:2.021982e+16 its
Lp:78.0227 & TOP Plane:A26.338311
Step:58254572 PN:1154047031 PN Depth:2.523008e+08 Sum:3.271636e+16 its
Lp:79.0227 & TOP Plane:A26.671644
Step:73629340 PN:1476842981 PN Depth:3.227960e+08 Sum:5.293618e+16 its
Lp:80.0227 & TOP Plane:A27.004978 *-12

Step:93069728 PN:1889788871 PN Depth:4.129459e+08 Sum:8.565255e+16 its
Lp:81.0227 & TOP Plane:A27.338311
Step:117652380 PN:2418027719 PN Depth:5.282388e+08 Sum:1.385887e+17 its
Lp:82.0227 & TOP Plane:A27.671644
Step:148739879 PN:3093685613 PN Depth:6.756579e+08 Sum:2.242413e+17 its
Lp:83.0227 & TOP Plane:A28.004978 *-9

Step:188056076 PN:3957867593 PN Depth:8.641820e+08 Sum:3.628300e+17 its
Lp:84.0227 & TOP Plane:A28.338311
Step:237782434 PN:5063171261 PN Depth:1.105304e+09 Sum:5.870713e+17 its
Lp:85.0227 & TOP Plane:A28.671644
Step:300679579 PN:6476687027 PN Depth:1.413516e+09 Sum:9.499013e+17 its
Lp:86.0227 & TOP Plane:A29.004978 *-6

Step:380241003 PN:8284239937 PN Depth:1.807553e+09 Sum:1.536973e+18 its
Lp:87.0227 & TOP Plane:A29.338311
Step:480888447 PN:10595705777 PN Depth:2.311466e+09 Sum:2.486874e+18 its
Lp:88.0227 & TOP Plane:A29.671644
Step:608217668 PN:13551217079 PN Depth:2.955511e+09 Sum:4.023847e+18 its
Lp:89.0227 & TOP Plane:A30.004978 *-3

Step:769312187 PN:17330081771 PN Depth:3.778865e+09 Sum:6.510721e+18 its
Lp:90.0227 & TOP Plane:A30.338311
Step:973137428 PN:22161454589 PN Depth:4.831373e+09 Sum:1.053457e+19 its
Lp:91.0227 & TOP Plane:A30.671644
Step:1231043204 PN:28338246133 PN Depth:6.176792e+09 Sum:1.704529e+19 its
Lp:92.0227 & TOP Plane:A31.004978 *

-end- Step:1299145937 Prime:29979245999 Sum:1.903105e+19 Lp:92.251689 idx:68
Base=27.000000 bias=0.000000




<IMG SRC="http://mister-computer.net/pixs/12DN-1175.jpg" ALIGN=right BORDER=5
WIDTH50% HEIGHT=50%>

This image on the right illustrates 12 inscribed pentagons which allows both
the 3 and 4 step Lp planes to be visualized as seen in the table's data.
It will be used to highlight the patterns found in the Table of Prime number
Sums

<CENTER><H3>Appendix: Volume and Scale </H3></CENTER>

Let a cubic centimeter [or cc = cm^3] be the standard volume unit. On our
human scale, this small sugar cube sized item seems small and ordinary.
What does Mercury's orbital volume of 8e23 cubic KM and the nuclear volume
have in common
with this standard cm^3 ?

Both are equally distant from a standard cubic centimeter;.
Imagine a cc as the mid zone between Nuclear and Planetary Orbit scales.
This means the volume of Mercury's orbit is as distant as the volume of a
nucleus when using a cc reference volume.

Compare it at whatever precision imaginable.
Pause to consider how planetary orbital volumes makes any human scale volume
seem insignificant;
whether its a cc or a cubic mile.

Can any basis of linearity be possible or even imaginable at these vast
ranges?

Surely VOLUME at the Nuclear scale would be a thing onto itself.
Normal spatial concepts of our macro scale would seem not apply over such a
scale distance.

Users of Nano technology are familiar with non-linear scale effects that
redefine our common macro sense of volume;
for example static electricity for both surfaces and shapes.

See the HOME Page for my formal thoughts on Space and Mass.

<CENTER><H4> LINKS <IMG SRC="../pixs/LINK-ARR.JPG" ></H4></CENTER>

author: RD O'Meara Oak Park, IL.

This WEB page address: "http://mister-computer.net/primesums/Primes3D.htm"
<A HREF="mailto:RDo.meara(a)mister-computer.net">Email of Author:
'RDo.meara(a)mister-computer.net'</A>


JID's SLOPE: <A HREF="jidslope.htm">The Universal Slope of Volume,
Mathematically and Physically.
&nbsp; AKA, the Rydberg constant of 1.0973~!</A>

5SPACE: <A HREF="./5space.htm">"Particle Mass mapped by (N/2)^5, N=1 to
22"</A>

My Real interests are in computer science, I have spent my career as a
cybernetic engineer/designer who
is concerned about the future of our digital data. Our entire cyber
infrastructure lacks "Data Sanity";
especially in the areas of forgery and security, for data and time stamps.

In that vein, I have designed a NEW multi-Computer SECURE Operating System,
aka <A HREF="http://mister-computer.net/13x5/13x5sum.htm">Posopip - a 3CPU
system</A>



HEDRONS : <A HREF="http://www.myspace.com/mister_computer"> A slide show of
'The new 1-2-3 Digital Method of Counting in Three Dimensions!</A>

<A HREF="http://www.myspace.com/mister_computer">5SPACE in MySpace</A>

22April2010


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

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