Final: An obvious pattern found in the first 37Million Prime Sums using the log of the golden ratio Lp!

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From: dlzc on 4 Mar 2010 16:39

---

Dear rom...:

On Mar 4, 1:54 pm, rom...(a)sbcglobal.net wrote:
> Subj: A regular pattern found in the first 37Million Prime
> Sums using the log of the golden ratio Lp!
....
> The Lp three step Pattern of the Prime Sums is clearly
> real and definitive;  its exact shape/structure remains
> unknown for now.

Perhaps something like:
http://www.huffingtonpost.com/huff-wires/20090514/us-odd-is-he-mad/images/0b31a3ad-e221-4b63-a440-0905483b7a49.jpg

David A. Smith

From: bill on 7 Mar 2010 17:18

---

On Mar 4, 12:54 pm, rom...(a)sbcglobal.net wrote:
> Subj: A regular pattern found in the first 37Million Prime Sums using the log
> of the golden ratio Lp!
>
> ------- Lp Results for the first 37 Million Primesums
>
> The appended Table reveals a pattern in the Primesums.
> It was found by summing the first 37 Million Primes
> and noting their sums based on Lp unit increments.  
>
> This P^1 or Lp +1 snapshot perspective reveals
> two more resonances at 3 and 4 Lp steps
> that seem to slowly change predictably.
>
> ---------------------
>
> The six column Table lists the step count,
> the Prime at that step: PN,  the PN/step ascending ratio,
> the sum of all Primes to that step  and its Golden Ratio log Lp,
> and the 3stepGRO of the Sums which very slowly climbs above 1.30901
>
> Also noted was the PN/Step ratio takes four Lp steps
> to increase by almost one.
>
> ---------------------
>
> For example, the Table's last three Lp+3 steps are:
>
> Step:8975619 PN:1.600215E8 PN/Step:17.8285 Sum:6.966776E14 its Lp:71.023491
>
>         ----PN/Step:18.0764 Sum:1.127248E15 its Lp:72.023491
>         PN/Step:18.3240 Sum:1.823926E15 its Lp:73.023492
>
> Step:18087624 PN:3.359332E8 PN/Step:18.5725 Sum:2.951174E15 its Lp:74.023492 &
> 3stepGRO 1.309016E0
>
>         PN/Step:18.8201 Sum:4.775100E15 its Lp:75.023492
>         ----PN/Step:19.0681 Sum:7.726275E15 its Lp:76.023492
>
> Step:36482222 PN:7.046670E8 PN/Step:19.3154 Sum:1.250137E16 its Lp:77.023492 &
> 3stepGRO 1.309017E0
>
>         Where 3stepGRO is 1.250137E16 / (1.250137E16 - 2.951174E15) = 1.309017~
>         and the PN/Step ratio is  7.0466E8 / 36482222  = 19..3154
>         and the unit four step example marked by "----"
>
> -----------------------
>
> 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.
>
> -------------------  Conclusions
>
> The Lp three step Pattern of the Prime Sums is clearly real and definitive;  
> its exact shape/structure remains unknown for now.
>
> The near unit PN/Step ratio of four Lp steps, also seems to predict an unknown
> for now.
>
> ------------------- Background
>
> 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 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(199) = 10.9999...,  
> Lp(pi) = 2.37885~,  Lp(e) = 2.078087~,  Lp(2) = 1.44042~.
>
> Px(4) = 6.854102~ , Lp(c) = 50.1313~
>
> Lucas numbers represent the integer steps of Px(n) series;
> 76, 123, 199 are the 9th, 10th and 11th Lucas numbers.
>
> ----------------------Contact Information
>
>  See my site for the latest material.
> <A HREF="http://mister-computer.net/index.htm">WEB Site"</A>
> RD OMeara Oak Park IL  4 Mar 2010
>
> -----------------------------------------TABLE
>
> Table:  1ST 37Million Prime Sums arranged by Lp, the log of the golden ratio!
>  4 March 2010 -  1.PrimeS...(a)Mister-Computer.net By RD OMeara @ 60302
>
> Step:25 PN:9.700000E1 PN/Step:0.0000

PN/step = 97/25 =4.04166666666666667 not 0.0000

> Sum:1.060000E3

Sum = 1058, not 1060

> its Lp:14.476004

How do you arrive at this value?

> 3stepGRO 0.000000E0

How do you get this value?

regards, Bill J

> Step:168 PN:9.970000E2 PN/Step:0.0000 Sum:7.612700E4 its Lp:23.358026 &
> 3stepGRO 0.000000E0
> Step:375 PN:2.557000E3 PN/Step:0.0000 Sum:4.411060E5 its Lp:27.008980 &
> 3stepGRO 0.000000E0
> Step:468 PN:3.323000E3 PN/Step:0.0000 Sum:7.141590E5 its Lp:28.010245 &
> 3stepGRO 0.000000E0
> Step:585 PN:4.261000E3 PN/Step:0.0000 Sum:1.158040E6 its Lp:29.014747 &
> 3stepGRO 0.000000E0
> Step:731 PN:5.527000E3 PN/Step:7.5609 Sum:1.875238E6 its Lp:30.016399 &
> 3stepGRO 1.307577E0
> Step:914 PN:7.129000E3 PN/Step:7.7998 Sum:3.034407E6 its Lp:31.016542 &
> 3stepGRO 1.307794E0
> Step:1143 PN:9.221000E3 PN/Step:8.0674 Sum:4.910652E6 its Lp:32.016913 &
> 3stepGRO 1.308595E0
>
> ---- 4 step pattern arrangement of the PN/Step ratio
>
> Step:1431 PN:1.193900E4 PN/Step:8.3431 Sum:7.950674E6 its Lp:33.018239 &
> 3stepGRO 1.308659E0
> Step:1792 PN:1.533100E4 PN/Step:8.5552 Sum:1.287299E7 its Lp:34.019618 &
> 3stepGRO 1.308418E0
> Step:2245 PN:1.984300E4 PN/Step:8.8388 Sum:2.083232E7 its Lp:35.019955 &
> 3stepGRO 1.308425E0
> Step:2814 PN:2.556100E4 PN/Step:9.0835 Sum:3.372014E7 its Lp:36.020740 &
> 3stepGRO 1.308530E0
>
> Step:3528 PN:3.291700E4 PN/Step:9.3302 Sum:5.458212E7 its Lp:37.021569 &
> 3stepGRO 1.308637E0
> Step:4425 PN:4.233100E4 PN/Step:9.5663 Sum:8.831903E7 its Lp:38.021647 &
> 3stepGRO 1.308687E0
> Step:5552 PN:5.455900E4 PN/Step:9.8269 Sum:1.429228E8 its Lp:39.021933 &
> 3stepGRO 1.308785E0
> Step:6969 PN:7.031300E4 PN/Step:10.0894 Sum:2.312757E8 its Lp:40.022128 &
> 3stepGRO 1.308908E0
>
> Step:8750 PN:9.037300E4 PN/Step:10.3283 Sum:3.742634E8 its Lp:41.022414 &
> 3stepGRO 1.308867E0
> Step:10990 PN:1.163410E5 PN/Step:10.5861 Sum:6.056061E8 its Lp:42.022534 &
> 3stepGRO 1.308899E0
> Step:13808 PN:1.495190E5 PN/Step:10.8284 Sum:9.800271E8 its Lp:43.022822 &
> 3stepGRO 1.308881E0
> Step:17353 PN:1.923070E5 PN/Step:11.0821 Sum:1.585836E9 its Lp:44.022979 &
> 3stepGRO 1.308906E0
>
> Step:21815 PN:2.472790E5 PN/Step:11.3353 Sum:2.566096E9 its Lp:45.023107 &
> 3stepGRO 1.308905E0
> Step:27430 PN:3.178570E5 PN/Step:11.5879 Sum:4.152187E9 its Lp:46.023185 &
> 3stepGRO 1.308946E0
> Step:34502 PN:4.086310E5 PN/Step:11.8437 Sum:6.718515E9 its Lp:47.023227 &
> 3stepGRO 1.308968E0
> Step:43406 PN:5.245190E5 PN/Step:12.0840 Sum:1.087127E10 its Lp:48.023321 &
> 3stepGRO 1.308975E0
>
> Step:54622 PN:6.741310E5 PN/Step:12.3417 Sum:1.759055E10 its Lp:49.023376 &
> 3stepGRO 1.308980E0
> Step:68751 PN:8.654090E5 PN/Step:12.5876 Sum:2.846254E10 its Lp:50.023406 &
> 3stepGRO 1.308982E0
> Step:86553 PN:1.111339E6 PN/Step:12.8400 Sum:4.605344E10 its Lp:51.023410 &
> 3stepGRO 1.308999E0
> Step:108990 PN:1.426933E6 PN/Step:13.0923 Sum:7.451630E10 its Lp:52.023418 &
> 3stepGRO 1.309008E0
>
> Step:137269 PN:1.831399E6 PN/Step:13.3417 Sum:1.205704E11 its Lp:53.023427 &
> 3stepGRO 1.309012E0
> Step:172925 PN:2.350631E6 PN/Step:13.5934 Sum:1.950889E11 its Lp:54.023446 &
> 3stepGRO 1.309010E0
> Step:217877 PN:3.015791E6 PN/Step:13.8417 Sum:3.156624E11 its Lp:55.023459 &
> 3stepGRO 1.309008E0
> Step:274568 PN:3.869111E6 PN/Step:14.0916 Sum:5.107534E11 its Lp:56.023463 &
> 3stepGRO 1.309010E0
>
> ---- 3 step pattern arrangement of the Lp(Sum)
>
>   Step:346068 PN:4.962677E6 PN/Step:14.3402 Sum:8.264175E11 its Lp:57.023466 &
> 3stepGRO 1.309013E0
>  Step:436266 PN:6.365773E6 PN/Step:14.5915 Sum:1.337176E12 its Lp:58.023472 &
> 3stepGRO 1.309014E0
> Step:550061 PN:8.164027E6 PN/Step:14.8420 Sum:2.163600E12 its Lp:59.023476 &
> 3stepGRO 1.309014E0
>
>   Step:693644 PN:1.046846E7 PN/Step:15.0920 Sum:3.500782E12 its Lp:60.023479 &
> 3stepGRO 1.309014E0
>  Step:874849 PN:1.342099E7 PN/Step:15.3409 Sum:5.664388E12 its Lp:61.023480 &
> 3stepGRO 1.309015E0
> Step:1103540 PN:1.720452E7 PN/Step:15.5903 Sum:9.165189E12 its Lp:62.023483 &
> 3stepGRO 1.309015E0
>
>   Step:1392226 PN:2.204914E7 PN/Step:15.8373 Sum:1.482960E13 its Lp:63.023486
> & 3stepGRO 1.309015E0
>  Step:1756660 PN:2.826113E7 PN/Step:16.0880 Sum:2.399480E13 its Lp:64.023486 &
> 3stepGRO 1.309015E0
> Step:2216797 PN:3.621858E7 PN/Step:16.3383 Sum:3.882444E13 its Lp:65.023488 &
> 3stepGRO 1.309016E0
>
>   Step:2797814 PN:4.640275E7 PN/Step:16.5854 Sum:6.281931E13 its Lp:66.023489
> & 3stepGRO 1.309016E0
>  Step:3531558 PN:5.945305E7 PN/Step:16.8348 Sum:1.016438E14 its Lp:67.023490 &
> 3stepGRO 1.309016E0
> Step:4458271 PN:7.615752E7 PN/Step:17.0823 Sum:1.644632E14 its Lp:68.023490 &
> 3stepGRO 1.309016E0
>
>   Step:5628807 PN:9.755560E7 PN/Step:17.3315 Sum:2.661071E14 its Lp:69.023491
> & 3stepGRO 1.309016E0
>  Step:7107499 PN:1.249508E8 PN/Step:17.5801 Sum:4.305703E14 its Lp:70.023491 &
> 3stepGRO 1.309016E0
> Step:8975619 PN:1.600215E8 PN/Step:17.8285 Sum:6.966776E14 its Lp:71.023491 &
> 3stepGRO 1.309016E0
>
>   Step:11335984 PN:2.049139E8 PN/Step:18.0764 Sum:1.127248E15 its Lp:72..023491
> & 3stepGRO 1.309016E0
>  Step:14318524 PN:2.623721E8 PN/Step:18.3240 Sum:1.823926E15 its Lp:73.023492
> & 3stepGRO 1.309016E0
> Step:18087624 PN:3.359332E8 PN/Step:18.5725 Sum:2.951174E15 its Lp:74.023492 &
> 3stepGRO 1.309016E0
>
>   Step:22851095 PN:4.300598E8 PN/Step:18.8201 Sum:4.775100E15 its Lp:75..023492
> & 3stepGRO 1.309016E0
>  Step:28871800 PN:5.505299E8 PN/Step:19.0681 Sum:7.726275E15 its Lp:76.023492
> & 3stepGRO 1.309017E0
> Step:36482222 PN:7.046670E8 PN/Step:19.3154 Sum:1.250137E16 its Lp:77.023492 &
> 3stepGRO 1.309017E0
>
> -end- step:37000000 Prime:7.152257E8 Sum:1.286897E16 Lp:77.083715 idx:53
> bias=26.0000
>
> - end of Table -
>
> --
>  Yours truly    RD   email mr.compu...(a)pobox.com

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