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From: rom126 on 19 Feb 2010 13:53
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 200K Primes

A regular pattern was found by summing the 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 five column table below lists the step count, the Prime at that step,
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.

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

For example, the last two selected P^3 steps are at 60,454 and 120,664 ;
they sum to 2.175057E10 cubics with a Lp of 49.464507, and to
9.213938E10 cubics with a Lp of 52.464563 and have a 3 step ratio of
120664/60454 = 1.996~.


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.


This spiral pattern will obviously go on indefinitely,
so the only question left concerns the 3 step ratio as the sums increase.

Does it continue to increase or will it remain below two?
I believe that it will never exceed two.

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

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.

RD OMeara Oak Park IL 19Feb2010

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

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

1ST 200k Prime Sums arranged by Lp, the log of the golden ratio!
17Feb2010 - All Rights Reserved by RD OMeara at
1.primes3d(a)mister-computer.net

step:25 Prime:9.700000E1 Sum:1.060000E3 Lp:14.476004
step:168 Prime:9.970000E2 Sum:7.612700E4 Lp:23.358026

step:516 Prime:3.697000E3 Sum:8.828890E5 Lp:28.450994

step:645 Prime:4.793000E3 Sum:1.430306E6 Lp:29.453555
step:806 Prime:6.199000E3 Sum:2.314985E6 Lp:30.454185 P^3-step:1.952
step:1008 Prime:8.009E3 Sum:3.746575E6 Lp:31.454656 P^3-step:1.953

step:1262 Prime:1.028900E4 Sum:6.071403E6 Lp:32.457848 P^3-step:1.957
step:1580 Prime:1.330900E4 Sum: 9.831005E6 Lp:33.459385 P^3-step:1.960
step:1979 Prime:1.718900E4 Sum:1.591132E7 Lp:34.459963 P^3-step:1.963

step:2480 Prime:2.212300E4 Sum:2.576657E7 Lp:35.461699 P^3-step:1.965
step:3109 Prime:2.857300E4 Sum:4.169913E7 Lp:36.462094 P^3-step:1.968
step:3899 Prime:3.677900E4 Sum:6.750125E7 Lp:37.463038 P^3-step:1.970

step:4891 Prime:4.743100E4 Sum:1.092210E8 Lp:38.463070 P^3-step:1.972
step:6138 Prime:6.091300E4 Sum:1.767459E8 Lp:39.463336 P^3-step:1.974
step:7705 Prime:7.851700E4 Sum:2.860221E8 Lp:40.463635 P^3-step:1.976

step:9676 Prime:1.010270E5 Sum:4.628404E8 Lp:41.463846 P^3-step:1.978
step:12154 Prime:1.299190E5 Sum:7.489364E8 Lp:42.463970 P^3-step:1.980
step:15273 Prime:1.671590E5 Sum:1.211905E9 Lp:43.464144 P^3-step:1.982

step:19196 Prime:2.147830E5 Sum:1.961011E9 Lp:44.464257 P^3-step:1.984
step:24134 Prime:2.761370E5 Sum:3.172990E9 Lp:45.464262 P^3-step:1.986
step:30351 Prime:3.546890E5 Sum:5.134306E9 Lp:46.464383 P^3-step:1.987

step:38178 Prime:4.561670E5 Sum:8.307560E9 Lp:47.464403 P^3-step:1.989
step:48036 Prime:5.858890E5 Sum:1.344217E10 Lp:48.464442 P^3-step:1.990
step:60454 Prime:7.528330E5 Sum:2.175057E10 Lp:49.464507 P^3-step:1.992

step:76101 Prime:9.664810E5 Sum:3.519356E10 Lp:50.464531 P^3-step:1.993
step:95815 Prime:1.241027E6 Sum:5.694489E10 Lp:51.464550 P^3-step:1.995
step:120664 Prime:1.593271E6 Sum:9.213938E10 Lp:52.464563 P^3-step:1.996

step:151985 Prime:2.043931E6 Sum:1.490847E11 Lp:53.464565 P^3-step:1.997
step:191475 Prime:2.623861E6 Sum:2.412249E11 Lp:54.464571 P^3-step:1.998

-end- cnt:200000 Prime:2.750159E6 Sum:2.641291E11 Lp:54.653071 idx:30
------- end table --------

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

From: Gerry Myerson on 22 Feb 2010 20:58
In article <phntn55ccfbjkvmittig7d4g206j1q1pgv(a)4ax.com>,
rom126(a)sbcglobal.net wrote:

> Assume Prime Numbers are Three Dimensional

If you really want to set off alarm bells that scream
CRANK POST COMING!!!!!
then the above is a great way to start.

I still can't make hide nor tail of your post,
but I do wonder whether anything you've done
really has anything to do with primes.
What happens if you repeat your calculations using,
say, all odd numbers instead of just the primes?
Or using the numbers [ n log n ], n = 1, 2, ...,
where [x] means the integer part of x
and the log in question is the natural log?
Come to think of it, what if you use natural logs
instead of golden ratio logs? Do you still get "patterns"?

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

More alarm bells.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: rom126 on 25 Feb 2010 09:56
On Tue, 23 Feb 2010 12:58:40 +1100, Gerry Myerson
<gerry(a)maths.mq.edi.ai.i2u4email> wrote:

>In article <phntn55ccfbjkvmittig7d4g206j1q1pgv(a)4ax.com>,
> rom126(a)sbcglobal.net wrote:
>
>> Assume Prime Numbers are Three Dimensional
>
>If you really want to set off alarm bells that scream
>CRANK POST COMING!!!!!
>then the above is a great way to start.
>
>I still can't make hide nor tail of your post,
>but I do wonder whether anything you've done
>really has anything to do with primes.
>What happens if you repeat your calculations using,
>say, all odd numbers instead of just the primes?
>Or using the numbers [ n log n ], n = 1, 2, ...,
>where [x] means the integer part of x
>and the log in question is the natural log?
>Come to think of it, what if you use natural logs
>instead of golden ratio logs? Do you still get "patterns"?
>
>> This 3D pattern was found by assuming Primes are cubic numbers
>> and that their sum represents the interior volume of the Prime spiral.
>
>More alarm bells.


Hey Gerry

Did you ever see any pattern in Primes Sums?

If so, pls enlighten me .

Don't you think the fact as the data support, that Prime Sums appear
to be 3D in a numerical sense, warrants some recognition?

I notice you did not find any errors in the data.

The paper states WHY Lp was used;
it has 1D, 2D and 3D aspects simultaneously!!!!

--
Regards from RD O'Meara

From: Gerry Myerson on 28 Feb 2010 21:04
In article <af3do5pkr7rmsab009jvsf5uvr1beg3ehc(a)4ax.com>,
rom126(a)sbcglobal.net wrote:

> On Tue, 23 Feb 2010 12:58:40 +1100, Gerry Myerson
> <gerry(a)maths.mq.edi.ai.i2u4email> wrote:
>
> >In article <phntn55ccfbjkvmittig7d4g206j1q1pgv(a)4ax.com>,
> > rom126(a)sbcglobal.net wrote:
> >
> >> Assume Prime Numbers are Three Dimensional
> >
> >If you really want to set off alarm bells that scream
> >CRANK POST COMING!!!!!
> >then the above is a great way to start.
> >
> >I still can't make hide nor tail of your post,
> >but I do wonder whether anything you've done
> >really has anything to do with primes.
> >What happens if you repeat your calculations using,
> >say, all odd numbers instead of just the primes?
> >Or using the numbers [ n log n ], n = 1, 2, ...,
> >where [x] means the integer part of x
> >and the log in question is the natural log?
> >Come to think of it, what if you use natural logs
> >instead of golden ratio logs? Do you still get "patterns"?
> >
> >> This 3D pattern was found by assuming Primes are cubic numbers
> >> and that their sum represents the interior volume of the Prime spiral.
> >
> >More alarm bells.
>
>
> Hey Gerry
>
> Did you ever see any pattern in Primes Sums?
>
> If so, pls enlighten me .

There are lots of patterns in the sum of the first n primes.
Any good text on analytic number theory will show you
how to estimate the sum of p, over all primes p less than x,
as a function of x.

> Don't you think the fact as the data support, that Prime Sums appear
> to be 3D in a numerical sense, warrants some recognition?

As I've mentioned several times, your presentation is impenetrable,
so it's not the least bit clear what your data is, much less what it may
or may not support.

> I notice you did not find any errors in the data.

It's hard to find errors when the presentation is so obscure.

> The paper states WHY Lp was used;
> it has 1D, 2D and 3D aspects simultaneously!!!!

You're missing the point. If some other log is used,
and if you still get the same patterns,
it will show that there's nothing special about
the golden ratio. That's the way mathematics works;
we try to falsify, not justify, our conclusions.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
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