mersenneforum.org twin prime count HOW very regularly?
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 2019-07-17, 22:05 #1 hal1se   Jul 2018 478 Posts 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. please think: 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
 2019-07-18, 17:40 #2 Dylan14     "Dylan" Mar 2017 20716 Posts 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 Tomas Oliveira e Silva, 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.
2019-07-19, 05:33   #3
CRGreathouse

Aug 2006

22·1,483 Posts

Quote:
 Originally Posted by Dylan14 And secondly, per Tomas Oliveira e Silva, 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.
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.

2019-07-19, 10:35   #4
hal1se

Jul 2018

3·13 Posts

Quote:
 Originally Posted by Dylan14 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 Tomas Oliveira e Silva, 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.
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.

Last fiddled with by hal1se on 2019-07-19 at 11:03

 2019-07-19, 13:31 #5 hal1se   Jul 2018 3·13 Posts https://alpertron.com.ar/ECM.HTM please paste: x=10**38+6*10**37+5*10**36;x=n(x);c<=1000;x please press 'only evaluate' button 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 very poor! please think: how can we calculate, faster 1e8 or may be 1e16 times.
 2019-08-02, 19:54 #6 hal1se   Jul 2018 3×13 Posts partial look and look: all if we wonder: twin system how like randomize, how different randomize? 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 44^8 times loop please. question: count=~? answer: count * 2,64 =~ twin prime count. question: twin prime system how different randomize test system? answer: if we look partial test result: 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: twin prime system how like randomize test system? answer: if we look all test result: twin system count near randomize test*2,64
2019-08-04, 18:12   #7
Dylan14

"Dylan"
Mar 2017

3·173 Posts

Quote:
 Originally Posted by hal1se if we wonder: twin system how like randomize, how different randomize? 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 44^8 times loop please. question: count=~? answer: count * 2,64 =~ twin prime count.

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)?
Attached Files
 hallsetwinproblem.7z (362.0 KB, 49 views)

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