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-   -   twin prime count HOW very regularly? (https://www.mersenneforum.org/showthread.php?t=24595)

 hal1se 2019-07-17 22:05

twin prime count HOW very regularly?

exponantial special ranges, prime and twin prime count: int[exp(n)]-n^8 to int[exp(n)]+n^8

n=32 : counts?
range: exp(32)-(32^8) to exp(32)+(32^8)
primesieve -q -c1 -c2 -c4 -c6 --time -s128 -t3 78962960182680-32**8 78962960182680+32**8
Primes: 68719534132
Twin primes: 2835414199

n=33 : counts?
primesieve -q -c1 -c2 -c4 -c6 --time -s128 -t3 214643579785916-33**8 214643579785916+33**8
Primes: 85236900427
Twin primes: 3410326616

question: twin prime count ? can it be estimated?
85236900427 / ((33*68719534132)/(32*2835414199))=3410358149
deviation:(3410358149-3410326616)/3410326616=9,2e-6

qbasic64 program for batch file:
OPEN "d:\ps\exp34to44.bat" FOR OUTPUT AS #1
DIM a AS _UNSIGNED _INTEGER64

PRINT #1, "echo ____" + "> exp34to44.txt"

FOR q = 34 TO 44
PRINT #1, "echo exp(" + MID\$(STR\$(q), 2) + ") >> exp34to44.txt"
a = INT(EXP(q))
PRINT #1, "primesieve -c1 -c2 -c4 -c6 --time -s128 -t4 -q " + STR\$(a) + "-" + MID\$(STR\$(q), 2) + "**8 ";
PRINT #1, STR\$(a) + "+" + MID\$(STR\$(q), 2) + "**8 " + ">> exp34to44.txt"

NEXT
CLOSE
END
REM end of file

n=34 to 44 batch file, for cmd command line:

echo ____> exp34to44.txt
echo exp(34) >> exp34to44.txt
primesieve -c1 -c2 -c4 -c6 --time -s128 -t3 -q 583461742527454-34**8 583461742527454+34**8 >> exp34to44.txt
echo exp(35) >> exp34to44.txt
primesieve -c1 -c2 -c4 -c6 --time -s128 -t3 -q 1586013452313430-35**8 1586013452313430+35**8 >> exp34to44.txt
echo exp(36) >> exp34to44.txt
primesieve -c1 -c2 -c4 -c6 --time -s128 -t3 -q 4311231547115195-36**8 4311231547115195+36**8 >> exp34to44.txt
echo exp(37) >> exp34to44.txt
primesieve -c1 -c2 -c4 -c6 --time -s128 -t3 -q 11719142372802612-37**8 11719142372802612+37**8 >> exp34to44.txt
echo exp(38) >> exp34to44.txt
primesieve -c1 -c2 -c4 -c6 --time -s128 -t3 -q 31855931757113756-38**8 31855931757113756+38**8 >> exp34to44.txt
echo exp(39) >> exp34to44.txt
primesieve -c1 -c2 -c4 -c6 --time -s128 -t3 -q 86593400423993744-39**8 86593400423993744+39**8 >> exp34to44.txt
echo exp(40) >> exp34to44.txt
primesieve -c1 -c2 -c4 -c6 --time -s128 -t3 -q 235385266837020000-40**8 235385266837020000+40**8 >> exp34to44.txt
echo exp(41) >> exp34to44.txt
primesieve -c1 -c2 -c4 -c6 --time -s128 -t3 -q 639843493530054912-41**8 639843493530054912+41**8 >> exp34to44.txt
echo exp(42) >> exp34to44.txt
primesieve -c1 -c2 -c4 -c6 --time -s128 -t3 -q 1739274941520500992-42**8 1739274941520500992+42**8 >> exp34to44.txt
echo exp(43) >> exp34to44.txt
primesieve -c1 -c2 -c4 -c6 --time -s128 -t3 -q 4727839468229346304-43**8 4727839468229346304+43**8 >> exp34to44.txt
echo exp(44) >> exp34to44.txt
primesieve -c1 -c2 -c4 -c6 --time -s128 -t3 -q 12851600114359308288-44**8 12851600114359308288+44**8 >> exp34to44.txt
REM end of file

n=34 : counts?
exp(34)
Seconds: 2880.734
Primes: 105046920323
Twin primes: 4079295626

question: twin prime count ? approximate value:
105046920323 / ((34*85236900427)/(33*3410326616))=4079309656
deviation:(4079309656-4079295626)/4079295626=3,4e-6

exp(35)
Seconds: 3600.814
Primes: 128678584218
Twin primes: 4854234505

question: twin prime count ? approximate value:
128678584218 / ((35*105046920323)/(34*4079295626))=4854214930
deviation:(4854214930-4854234505)/4854234505=-4,03e-6

exp(36)
Seconds: 4343.392
Primes: 156728004566
Twin primes: 5748119658
156728004566 / ((36*128678584218)/(35*4854234505))=5748130599
dev:(5748130599-5748119658)/5748119658=1,9e-6

exp(37)
Seconds: 5647.871
Primes: 189863714848
Twin primes: 6775131690
189863714848 / ((37*156728004566)/(36*5748119658))=6775197297
dev:(6775197297-6775131690)/6775131690=9,7e-6

exp(38)
Seconds: 7180.391
Primes: 228831310050
Twin primes: 7950932791
228831310050 / ((38*189863714848)/(37*6775131690))=7950772767
dev:(7950772767-7950932791)/7950932791=-2,0e-5
abs(dev)>1e-5 but 2/(10**5) mini value.

exp(39)
Seconds: 9078.952
Primes: 274462036937
Twin primes: 9291718004
274462036937 / ((39*228831310050)/(38*7950932791))=9291886734
dev:(9291886734-9291718004)/9291718004=1,8e-5

exp(40)
Seconds: 12329.989
Primes: 327680132730
Twin primes: 10816086221
327680132730 / ((40*274462036937)/(39*9291718004))=10816044496
dev:(10816044496-10816086221)/10816086221=-3,9e-6

exp(41)
Seconds: 14655.309
Primes: 389508780389
Twin primes: 12543315601
389508780389 / ((41*327680132730)/(40*10816086221))=12543346411
dev:(12543346411-12543315601)/12543315601=2,5e-6

exp(42)
Seconds: 20925.439
Primes: 461078345073
Twin primes: 14494524702
461078345073 / ((42*389508780389)/(41*12543315601))=14494538414
dev:(14494538414-14494524702)/14494524702=9,5e-7

exp(43)
Seconds: 29125.672
Primes: 543637516493
Twin primes: 16692556020
543637516493 / ((43*461078345073)/(42*14494524702))=16692427846
dev:(16692427846-16692556020)/16692556020=-7,7e-6

exp(44)
Seconds: 39701.873
Primes: 638556198014
Twin primes: 19161347348
638556198014 / ((44*543637516493)/(43*16692556020))=19161448061
dev:(19161448061-19161347348)/19161347348=5,3e-6

question: if we have only twin count information, how can we find other twin counts?
exp(39) twin count = 9291718004 then exp(42) twin count approximate how?
9291718004*42^6/39^6=14494529452
dev:(14494529452-14494524702)/14494524702=3,3e-7
another test: exp(32) twin count =2835414199 then exp(41) twin count approximate how?
2835414199*41^6/32^6=12543530214
dev:(12543530214-12543315601)/12543315601=1,7e-5
another test: exp(40) twin count=10816086221 then exp(37) twin count approximate how?
10816086221*37^6/40^6=6775175307
dev:(6775175307-6775131690)/6775131690=6,4e-6
another test: exp(43) twin count=16692556020 then exp(36) twin count approximate how?
16692556020 * 36^6/43^6=5748137040
dev:(5748137040-5748119658)/5748119658=3,0e-6
another test:exp(42) twin count=14494524702 then exp(88) twin count approximate how?
14494524702*88^6/42^6=1226321293360
another test:exp(44) twin count=19161347348 then exp(88) twin count approximate how?
19161347348*88^6/44^6=19161347348*2^6=1226326230272
two different approximate value. these values very near.

if can you test exp(88)-88^8 to exp(88)+88^8 twin prime real count, you must see: deviation < 1e-4=1/(10**4)

dear programmer, please make twin prime count for range exp(45) to exp(88)
if you wonder and try upper values, you must see: abs(deviation) < 1e-4=1/(10**4)

exp(big) range, for example exp((10**12)**(10**12)) ranges HOW regularly?

i feel, for big ranges regularly without calculation.

question: twin prime count HOW very regularly?
i am an autistic, i love number regularities.
please forgive my words many mistake and not good fluent.
my brain damage.
disavantage: no!
may be avantage.
we look full picture, sometimes.

a few tips: for HOW question.

in the special range:exp(n)-n^8 to exp(n)+n^8,
twin count compare: near other many due prime system:

near: cousin prime count,

near: sophie germain p, 2p+1 due prime's first prime count =~ twin prime count. sophie prime count %6 or % 8 bigger then twin prime count in every big exp ranges. fluctation %2,5(not:only p in the range),

near: G=2*int[int[exp(n)] /6 ]*6 symetric goldbach due prime count (p+q=G, p and q symmetric all primes on point G/2,p and q in the range),

near: G=2*int[int[exp(n)] /6 ]*6+2 symetric goldbach due prime count *2,

near: G=2*int[int[exp(n)] /6 ]*6+4 symetric goldbach due prime count *2,
so: (G mod 6=0 symmetric primes count) =~ (G mod 6=2 symmetric primes count)+(G mod 6=4 symmetric primes count)
, this mean =~ : not exatly equal, % 10 fluctational!

near: please select (n^8) times 2 randomize integer in the range and look: these two integer same time prime then count=count+1, randomize count*(2,64...) near twin count,

near: please mixed 2*n^8 sequantial integer: in the range: exp(n)-n^8 to exp(n)+n^8, mixed and mixed. and select two integer sequantial. these two integer same time prime then count=count+1,randomize count*(2,64...) near twin count, so posible come back randomize or not come back randomize: not important!
this 2,64... a fix value, every big exponantial ranges! randomize due count and twin count rate: allways a fix value every big exponantial range: 2,64...

so: prime system regular base randomize, so:axiomatic, so:predicitive, so:formulative.
randomize: not gambling! if we look many big randomize integers, these type systems predictive.

math very easy, if think simple, and step by step.
---
end of text

 Dylan14 2019-07-18 17:40

Firstly, a suggestion: you may want to put your results in a table (in a pdf document, for instance) to improve readability (as it stands, the post is quite long and people won’t want to read all of it).
And secondly, per [URL="http://sweet.ua.pt/tos/primes.html#t2"]Tomas Oliveira e Silva[/URL], the number of twin primes have been calculated to at least 4*10^18 (about exp(42.833)). You want a twin prime count near exp(88), which is about 1.65*10^38. Yeah, I don’t foresee a calculation of that being feasible anytime soon.

 CRGreathouse 2019-07-19 05:33

[QUOTE=Dylan14;521871]And secondly, per [URL="http://sweet.ua.pt/tos/primes.html#t2"]Tomas Oliveira e Silva[/URL], the number of twin primes have been calculated to at least 4*10^18 (about exp(42.833)). You want a twin prime count near exp(88), which is about 1.65*10^38. Yeah, I don’t foresee a calculation of that being feasible anytime soon.[/QUOTE]

As a ballpark figure, Oliveira e Silva's calculation probably took in the neighborhood of \$10,000 to \$100,000 of compute time. Counting twin primes scales, at best, as x/log log x (practically speaking, it's superlinear) so ~ 3e23 dollars or ~ 4 billion years worth of gross world product. Knock off a couple of zeros for efficiency improvements and a few more for as many generations of Moore's law as you're willing to count on.

 hal1se 2019-07-19 10:35

[QUOTE=Dylan14;521871]Firstly, a suggestion: you may want to put your results in a table (in a pdf document, for instance) to improve readability (as it stands, the post is quite long and people won’t want to read all of it).
And secondly, per [URL="http://sweet.ua.pt/tos/primes.html#t2"]Tomas Oliveira e Silva[/URL], the number of twin primes have been calculated to at least 4*10^18 (about exp(42.833)). You want a twin prime count near exp(88), which is about 1.65*10^38. Yeah, I don’t foresee a calculation of that being feasible anytime soon.[/QUOTE]

30k{11,13;17,19;29,31}
so: every 30 integer only 6 prime test.
or
210k{...}
(7-2)*(5-2)*(3-2)=15 probably twin
so: 2*15/210
every 210 integer only 30 prime test:1/7,
30/210 * (2*88^8)=1,02752721373008e+15 primalize test

exp(88)=~1,65e+38

39 digit primalize test only 3 milisecond (poor technic 2019), so: every second 333 primalize test.
if we have 10e6 parellel processor then:

each proc. only 2*88^8/10e6=719269049 integer so only 719269049/7 =102752721 primalize test.
1,02752721373008e+15 /10e6/ 333/ 3600=85,7 hours

if we have 1e6 parellel processor then:
857 hours or 35 days

if you can have optimal technic (so:not use poor technic) only a few femto seconds need.

math must more groving for primalize test.

 hal1se 2019-07-19 13:31

[url]https://alpertron.com.ar/ECM.HTM[/url]
x=10**38+6*10**37+5*10**36;x=n(x);c<=1000;x
999 prime search only 2,5 or 3 seconds my very old AMD laptop. (chrome browser faster than other browser, about %250 fast)
so every seconds average 333 prime:ok

but this technic [B]very poor![/B]

please think: how can we calculate, faster 1e8 or may be 1e16 times.

 hal1se 2019-08-02 19:54

partial look and look: all

if we wonder: [B]twin system how like randomize, how different randomize?[/B]

please take two randomize integer in this range:

int(exp(44))-44^8 to int(exp(44))+44^8.

and look: two integer, same time primes than count=count+1

question: count=~?

answer: count * 2,64 =~ twin prime count.

question:[B] twin prime system how different randomize test system?[/B]

answer:[B] if we look partial test result[/B]: twin system different randomize test?

for example: randomize test may be sometimes: 1e5 times no appear same time two prime:

but twin system: sequantial 1e5 integer no twin imposible,

because: maximal twin gap < 44*44*44/1,32032=~64518

ln( exp(44)+44^8 )=44,00000109311=~44

so, if we look all integers in the range: twin gap > 65000 integers imposible!

but 100000 sequantial randomize, same time no appear two prime may be posible, sometimes!

quesion: [B]twin prime system how like randomize test system?[/B]

answer: [B]if we look all test result[/B]: twin system count near randomize test*2,64

 Dylan14 2019-08-04 18:12

1 Attachment(s)
[QUOTE=hal1se;522949]if we wonder: [B]twin system how like randomize, how different randomize?[/B]

please take two randomize integer in this range:

int(exp(44))-44^8 to int(exp(44))+44^8.

and look: two integer, same time primes than count=count+1

question: count=~?

answer: count * 2,64 =~ twin prime count.

[/QUOTE]

Assuming I understand you correctly, I created a Mathematica notebook to test this out (see attachment). However, due to timing constraints (I came to an estimate of 9.7 years to run the entire program), I ran the loop for 100 million iterations and then extrapolated to get the twin count. I then compared that to primesieve 7.4 and the number I got was less than 1% off the primesieve value. So in this case your idea makes sense. Some questions I have though:
1. Does this work for larger n than 44?
2. Continuing on this question, what is the asymptotic behavior of doing this if we replace 44 with n and let n go to infinity? Do we get the actual twin prime count, or does it diverge (and if so, does it grow or shrink relative to the actual value)?

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