Subj: A 3D pattern found in Prime Number sums - using the golden ratio log!
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From: rom126 on 17 Feb 2010 21:35 Subj: A 3D pattern found in Prime Number sums - using the golden ratio log! I have found a pattern in prime sums by assuming Prime Numbers are volumetric and examining them using Lp, the log of the golden ratio (sqrt(5) + 1) / 2 . In golden ratio terms, 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. The pattern was found by summing the primes and noting sums based on Lp increments of one, that reveal a volumetric pattern documented in the table below. The five column table lists the step count, the Prime at that step, the sum of all Primes to that step, the log of the golden ratio- Lp, and the plus 3 Lp volume step ratio which is approximately two. Every third line is prefixed with an asterisk to signal sequential P^3 steps. For example, the first 120,664 primes sum to 9.213938E10, with a Lp of 52.464563 and 3 log step ratio of 120664/60454 = 1.996~. This spiral pattern will obviously go on indefinitly,so the only question left concerns the 3 step ratio at large prime sum steps. 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. --------------- See my Web site for this and other discoveries http://mister-computer.net/index.htm -- Regards from RD OMeara Oak Park IL , 17Feb2010 ----------- 17Feb2010 Table of Prime sums at Golden Ratio Logs --------- 1ST 200k Prime Sums arranged by Lp, the log of the golden ratio! All Rights Reserved - by RD OMeara at 1.primes3d(a)mister-computer.net step:25 Prime:9.700000E1 Sum:1.060000E3 Lp:14.476004 st ep:168 Prime:9.970000E2 Sum:7.612700E4 Lp:23.358026 step:413 Prime:2.843000E3 Sum:5.441840E5 Lp:27.445383 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:8.009000E3 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 bias=26.4400 ------- end table -------- -- Yours truly RD email mr.computer(a)pobox.com
From: Gerry Myerson on 17 Feb 2010 23:38 In article <nr9pn5ph2blkqbsbjl6gfuu68h4dr6ckkj(a)4ax.com>, rom126(a)sbcglobal.net wrote: > Subj: A 3D pattern found in Prime Number sums - using the golden ratio log! > > > I have found a pattern in prime sums by assuming Prime Numbers are volumetric > and examining them using Lp, the log of the golden ratio (sqrt(5) + 1) / 2 . So Lp is a number, the log (to some unspecified base) of the golden ratio. But then somehow you keep getting different values of Lp, from 14.47 to 23.35 to ... to 54.65, so evidently Lp is NOT the log of the golden ratio. So what is it really? And what is a P^3 step? > In golden ratio terms, 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. > > The pattern was found by summing the primes and > noting sums based on Lp increments of one, that > reveal a volumetric pattern documented in the table below. But the Lp increments aren't one, they're just close to one. I'm sorry, none of this makes any sense to me. Can you write down formulas for any of this? -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: rom126 on 19 Feb 2010 09:42
On Thu, 18 Feb 2010 15:38:25 +1100, Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> wrote: >In article <nr9pn5ph2blkqbsbjl6gfuu68h4dr6ckkj(a)4ax.com>, > rom126(a)sbcglobal.net wrote: > >> Subj: A 3D pattern found in Prime Number sums - using the golden ratio log! >> >> >> I have found a pattern in prime sums by assuming Prime Numbers are volumetric >> and examining them using Lp, the log of the golden ratio (sqrt(5) + 1) / 2 . > >So Lp is a number, the log (to some unspecified base) of >the golden ratio. > >But then somehow you keep getting different values of Lp, >from 14.47 to 23.35 to ... to 54.65, >so evidently Lp is NOT the log of the golden ratio. >So what is it really? > >And what is a P^3 step? > >> In golden ratio terms, 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. >> >> The pattern was found by summing the primes and >> noting sums based on Lp increments of one, that >> reveal a volumetric pattern documented in the table below. > >But the Lp increments aren't one, they're just close to one. > >I'm sorry, none of this makes any sense to me. Can you write >down formulas for any of this? Thanks for the comments Gerry. Lp is the log to base 1.618.... Lp(X) = Ln( X ) / Ln(1.618....) . for example Lp(123) = 10.000137.. ; and the inverse Px(10) = 122.991.... Lp(199) = 10.9999..... Lp(pi) = 2.37885~, Lp(e) = 2.078087~ Look up Lucas numbers, they are integer steps of Px(n) series. -------- 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. ------------------------------- As the Primes accumulate their assumed cubic sums, they also follow a seemingly irregular yet discrete growth pattern as shown by the P^3 step ratio approaching two. 120664/60454 = 1.996~ step: 60454 Prime:7.528330E5 Sum:2.175057E10 Lp:49.464507 P^3-step:1.992 step:120664 Prime:1.593271E6 Sum:9.213938E10 Lp:52.464563 P^3-step:1.996 All that is revealed is: Prime Numbers are cubic in nature and their sums appear to fill up an unknown volume; whose exact geometry has yet to be seen. The web site will always have the latest info, so check there also. http://mister-computer.net/index.htm -- Regards from RD -- Yours truly RD email mr.computer(a)pobox.com |