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Next: Final: An obvious pattern found in the first 37Million Prime Sums using the log of the golden ratio Lp!
From: rom126 on 4 Mar 2010 15:54
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.PrimeSums(a)Mister-Computer.net By RD OMeara @ 60302

Step:25 PN:9.700000E1 PN/Step:0.0000 Sum:1.060000E3 its Lp:14.476004 &
3stepGRO 0.000000E0
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.computer(a)pobox.com

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