20190717, 22:05  #1 
Jul 2018
2^{3}×5 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 7896296018268032**8 78962960182680+32**8 Primes: 68719534132 Twin primes: 2835414199 n=33 : counts? primesieve q c1 c2 c4 c6 time s128 t3 21464357978591633**8 214643579785916+33**8 Primes: 85236900427 Twin primes: 3410326616 question: twin prime count ? can it be estimated? 85236900427 / ((33*68719534132)/(32*2835414199))=3410358149 deviation:(34103581493410326616)/3410326616=9,2e6 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 58346174252745434**8 583461742527454+34**8 >> exp34to44.txt echo exp(35) >> exp34to44.txt primesieve c1 c2 c4 c6 time s128 t3 q 158601345231343035**8 1586013452313430+35**8 >> exp34to44.txt echo exp(36) >> exp34to44.txt primesieve c1 c2 c4 c6 time s128 t3 q 431123154711519536**8 4311231547115195+36**8 >> exp34to44.txt echo exp(37) >> exp34to44.txt primesieve c1 c2 c4 c6 time s128 t3 q 1171914237280261237**8 11719142372802612+37**8 >> exp34to44.txt echo exp(38) >> exp34to44.txt primesieve c1 c2 c4 c6 time s128 t3 q 3185593175711375638**8 31855931757113756+38**8 >> exp34to44.txt echo exp(39) >> exp34to44.txt primesieve c1 c2 c4 c6 time s128 t3 q 8659340042399374439**8 86593400423993744+39**8 >> exp34to44.txt echo exp(40) >> exp34to44.txt primesieve c1 c2 c4 c6 time s128 t3 q 23538526683702000040**8 235385266837020000+40**8 >> exp34to44.txt echo exp(41) >> exp34to44.txt primesieve c1 c2 c4 c6 time s128 t3 q 63984349353005491241**8 639843493530054912+41**8 >> exp34to44.txt echo exp(42) >> exp34to44.txt primesieve c1 c2 c4 c6 time s128 t3 q 173927494152050099242**8 1739274941520500992+42**8 >> exp34to44.txt echo exp(43) >> exp34to44.txt primesieve c1 c2 c4 c6 time s128 t3 q 472783946822934630443**8 4727839468229346304+43**8 >> exp34to44.txt echo exp(44) >> exp34to44.txt primesieve c1 c2 c4 c6 time s128 t3 q 1285160011435930828844**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:(40793096564079295626)/4079295626=3,4e6 exp(35) Seconds: 3600.814 Primes: 128678584218 Twin primes: 4854234505 question: twin prime count ? approximate value: 128678584218 / ((35*105046920323)/(34*4079295626))=4854214930 deviation:(48542149304854234505)/4854234505=4,03e6 exp(36) Seconds: 4343.392 Primes: 156728004566 Twin primes: 5748119658 156728004566 / ((36*128678584218)/(35*4854234505))=5748130599 dev:(57481305995748119658)/5748119658=1,9e6 exp(37) Seconds: 5647.871 Primes: 189863714848 Twin primes: 6775131690 189863714848 / ((37*156728004566)/(36*5748119658))=6775197297 dev:(67751972976775131690)/6775131690=9,7e6 exp(38) Seconds: 7180.391 Primes: 228831310050 Twin primes: 7950932791 228831310050 / ((38*189863714848)/(37*6775131690))=7950772767 dev:(79507727677950932791)/7950932791=2,0e5 abs(dev)>1e5 but 2/(10**5) mini value. exp(39) Seconds: 9078.952 Primes: 274462036937 Twin primes: 9291718004 274462036937 / ((39*228831310050)/(38*7950932791))=9291886734 dev:(92918867349291718004)/9291718004=1,8e5 exp(40) Seconds: 12329.989 Primes: 327680132730 Twin primes: 10816086221 327680132730 / ((40*274462036937)/(39*9291718004))=10816044496 dev:(1081604449610816086221)/10816086221=3,9e6 exp(41) Seconds: 14655.309 Primes: 389508780389 Twin primes: 12543315601 389508780389 / ((41*327680132730)/(40*10816086221))=12543346411 dev:(1254334641112543315601)/12543315601=2,5e6 exp(42) Seconds: 20925.439 Primes: 461078345073 Twin primes: 14494524702 461078345073 / ((42*389508780389)/(41*12543315601))=14494538414 dev:(1449453841414494524702)/14494524702=9,5e7 exp(43) Seconds: 29125.672 Primes: 543637516493 Twin primes: 16692556020 543637516493 / ((43*461078345073)/(42*14494524702))=16692427846 dev:(1669242784616692556020)/16692556020=7,7e6 exp(44) Seconds: 39701.873 Primes: 638556198014 Twin primes: 19161347348 638556198014 / ((44*543637516493)/(43*16692556020))=19161448061 dev:(1916144806119161347348)/19161347348=5,3e6 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:(1449452945214494524702)/14494524702=3,3e7 another test: exp(32) twin count =2835414199 then exp(41) twin count approximate how? 2835414199*41^6/32^6=12543530214 dev:(1254353021412543315601)/12543315601=1,7e5 another test: exp(40) twin count=10816086221 then exp(37) twin count approximate how? 10816086221*37^6/40^6=6775175307 dev:(67751753076775131690)/6775131690=6,4e6 another test: exp(43) twin count=16692556020 then exp(36) twin count approximate how? 16692556020 * 36^6/43^6=5748137040 dev:(57481370405748119658)/5748119658=3,0e6 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 < 1e4=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) < 1e4=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 
20190718, 17:40  #2 
"Dylan"
Mar 2017
2×3^{3}×11 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. 
20190719, 05:33  #3  
Aug 2006
5,987 Posts 
Quote:


20190719, 10:35  #4  
Jul 2018
2^{3}·5 Posts 
Quote:
so: every 30 integer only 6 prime test. or 210k{...} (72)*(52)*(32)=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 20190719 at 11:03 

20190719, 13:31  #5 
Jul 2018
2^{3}·5 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. 
20190802, 19:54  #6 
Jul 2018
2^{3}·5 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 
20190804, 18:12  #7  
"Dylan"
Mar 2017
2·3^{3}·11 Posts 
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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