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 2018-07-27, 10:27 #1 hal1se   Jul 2018 1001112 Posts small prime gap, regularly? primes: p1 < p2 < p3 prime gap: p2 - p1 or another gap: p3 - p2, but not p3 - p1 small prime gap: 2,4,6,...,18,20 analysis: g2:twin, g4:cousin, etc.. int exp(N) middle point. 100,000 primes range, fix! int exp(N) -+ 50000 primes N (range first prime) (range last prime) 53 283075330327469387402809 283075330327469392764067 count p g2 g4 g6 g8 g10 g12 g14 g16 g18 g20 100000 2425 2419 4652 2253 2896 4220 2404 1976 3841 2417 range gap rate: 100000/53=1886,7924528: prime count / ln(middle point of range) g2=2425 =~ range gap rate * 4/3 g4=~ range gap rate *g2 =~ range gap rate * 4/3 g6=4652 =~ range gap rate * 5/2 g8=2253 =~ range gap rate * 6/5 g10=2896 =~ range gap rate * 3/2 g12=4220 =~ range gap rate * 9/4 g14=2404 =~ range gap rate * 5/4 g16=1976 =~ range gap rate * 21/20 g18=3841 =~ range gap rate * 2 g20=2417 =~ range gap rate * 9/7 question: rational number fractions: 4/3, 5/2, ... , roughly expressed, are they provided on a regular basis? N (range first prime) (range last prime) 44 12851600114357107349 12851600114361502717 count p g2 g4 g6 g8 g10 g12 g14 g16 g18 g20 100000 3048 3031 5696 2619 3596 5078 2973 2312 4504 2662 ln(middle point of range)=~N rational number fractions: 4/3, 5/2, ? 1e5/44 = 2272,727 range gap rate=rg rg*4/3=~3030 very near g2:3048 or g4:3031 , g2,g4:ok, g2=~g4:so, twin count =~ cousin count.. rg*5/2=~5681 very near g6:5696 rg*6/5=~2727 near. g8:2619 rg*3/2=~3409 near g10:3596 rg*9/4=~5113 very near g12:5078 rg*5/4=~2840 near g14:2973 rg*21/20=~2386 very near g16:2312 rg*2=~4545 very near g18:4504 rg*9/7=~2922 not near g20:2622 g2,g4,g12,g16,g18 probably regular basis? test: exp(90) + 14000 primes: 14e3 primes range. exp(90)=~1220403294317840802002710035136369753970,746... middle point:=~int exp 90=1220403294317840802002710035136369753970 middle point +14000 primes: 90 1220403294317840802002710035136369754051 1220403294317840802002710035136371030577 count p g2 g4 g6 g8 g10 g12 g14 g16 g18 g20 14000 198 203 403 206 251 378 212 188 353 232 14000/90=155,55556:range gap rate. g2,g4,g12,g16,g18 probably regular basis? rg*4/3=207 very near g2=198, g4=203 rg*9/4=350 near g12=378 rg*21/20=163 near g16=188 rg*2=311 near g18=353 rg*5/2=388 near g6:403 rg*6/5=186 near g8:206 rg*3/2=233 near g10=251 rg*5/4=194 near g14=212 rg*9/7=200 not near g20:232 (very near) only g2,g4 middle point + 50000 90 1220403294317840802002710035136369754051 1220403294317840802002710035136374284647 count p g2 g4 g6 g8 g10 g12 g14 g16 g18 g20 50000 742 724 1442 727 905 1324 798 666 1275 800 range gap rate:rg=50000/90=555,5556 rg*4/3=740 very near g2:742, near g4:724 rg*9/4=1666 not near g12:1442 rg*21/20=583 not near g16:666 rg*2=1111 not near 1275 rg*5/2=1388 near g6:1442 rg*6/5=666 not near g8:727 rg*5/4=694 not near g14:798 rg*9/7=714 not near g20:800 very simple test: look at please: 3 ranges for 1e5 (or 5e4) primes. prime gaps if g>4 then, very complex and not regularly. only g2:twin and g4:cousin prime gap very regularly. i tested many ranges: twin prime count =~ cousin prime count and regularly every range! if: middle point of range > exp(13) and if: ln(middle point of range)/ln(range) < exp(1) than lower limit twin or cousin prime count: (4/3) range / ( (ln(middle point of range))^2 ) good by number lowers.
 2018-07-28, 17:50 #2 hal1se   Jul 2018 1001112 Posts how is it, small prime gaps, g2,g4 regularly, but g6,g8,... not? if any math model regularly then, we must see in every optimal ranges! i tested prime small gap again: 2,4,6,...,20 ranges: exp(14) to exp(44) -+ about 50e3 primes about 100 000 primes, 35 ranges: prime gap rate: prime count of range/ln(middle point of range) g6 gap rate:2,23 to 2,53 growing! if exp(90) : g6 up to 2,57: more growing! g8,g10, ... other rates: growing! prime gaps: if g>4 then not regularly! only g2:twin and g4:cousin gap rate, fluctuating but regularly! how is it? if number >2 then prime template: (2-1)=1, so only 1 combination. 2N+{1} if number >6 then prime template: (2-1)(3-1)=2, so only 2 combination. 6N+{1,5} 1 proably cousin = 1 probably twin if number >30 then prime template (2-1)(3-1)(5-1)=8, so only 8 combination. 30N+{+1,+7,+11,+13,+17,+19,+23,+29} {+31,...} ..........c...t...c...t....c.......t c:probably cousin t:probably twin 3=3 equal! if number >210 then prime template (2-1)(3-1)(5-1)(7-1)=48, so only 48 combination. 210N+{1,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,121,127,131,137,139,143,149,151,157,163,167,169,173,179,181,187,191,193,197,199,209} important:143,187 not prime, and in the template! probably cousin=18 probaly twin=15 18>15 but near! if number >2310 then prime template (2-1)(3-1)(5-1)(7-1)(11-1)=480, only 480 combination: 2310N+{1,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,169,173,179,181,191,193,197,199,211,...,2291,2293,2297,2309} +1,+2309 first +1:fix! last:last element-1 template 3#: 2,3 elimination template 5#: 2,3,5 elimination template 7#: 2,3,5,7 elimination tepmlate 11#:2310: 2,3,5,7,11 elimination and 143,187 elimnation, because: 143 mod 11=187 mod 11=0 temlate 11# probably cousin, twin ? i don't not, but some one count it please! if number > 30030 then (2-1)(3-1)(5-1)(7-1)(11-1)(13-1) = 5760 , only 5760 combination! 30030N+{1,17,...,30029} :{inculede 5760 element} probably twin <=>? probaly cousin i don't know! but very near. some one count please! abs (probaly cousin - probably twin) <210 210/5760<%3,6 so abs(twin prime count-cousin prime count) < %4. very simple! how is it, g6 not regularly? very simple: 5#=30 template: 30N+{+1,+7,+11,+13,+17,+19,+23,+29} ......g6......................g6 direct probably g6 only 2 times! but +13 elemnt not prime and +11,+17 elements are primes then extra g6 and: +17:prime, +19 not prime, +23 prime than extra second g6. if template 7#: extra g6 growing! if template 11#: extra g6 more growing! but g2, g4 not growing, only smal fluctuations,very regularly. because, every g2 and every g4 direct element, not inside extra g2, g4. direct elements g2,g4 fix and countable! very simple, if look full picture! ____________________ middle point:int exp( 14 ) 502613 to 1902569 prime count: 100471 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 9673 9573 16019 6735 8543 10184 5286 3765 6581 3153 prime gap rates: for exp( 14 )-+ 100471 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.347872 1.333937 2.232147 .9384798 1.190413 1.419076 .7365708 .524629 .9170209 .4393507 _______________________ middle point:int exp( 15 ) 2519017 to 4019009 prime count: 100049 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 8931 8831 15107 6315 8209 9845 5370 3803 6630 3320 prime gap rates: for exp( 15 )-+ 100049 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.338994 1.324001 2.26494 .946786 1.230747 1.476027 .8051055 .5701706 .994013 .4977561 _______________________ middle point:int exp( 16 ) 8086129 to 9686093 prime count: 100048 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 8316 8255 14193 6103 7941 9633 5375 3747 6646 3457 prime gap rates: for exp( 16 )-+ 100048 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.329922 1.320166 2.26979 .9760115 1.26995 1.540541 .8595874 .5992324 1.06285 .5528547 _______________________ middle point:int exp( 17 ) 23304959 to 25004939 prime count: 100005 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 7826 7850 13476 5904 7548 9418 5022 3757 6674 3535 prime gap rates: for exp( 17 )-+ 100005 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.330353 1.334433 2.290806 1.00363 1.283096 1.60098 .8536973 .638658 1.134523 .60092 _______________________ middle point:int exp( 18 ) 64759991 to 66559957 prime count: 100094 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 7313 7377 12950 5679 7257 9177 5003 3759 6720 3556 prime gap rates: for exp( 18 )-+ 100094 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.315104 1.326613 2.328811 1.02126 1.305033 1.650309 .8996943 .6759846 1.208464 .6394789 _______________________ middle point:int exp( 19 ) 177532339 to 179432299 prime count: 100049 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 6921 6903 12391 5406 7055 8889 4952 3689 6711 3458 prime gap rates: for exp( 19 )-+ 100049 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.314346 1.310928 2.353137 1.026637 1.339793 1.688083 .9404192 .7005667 1.274466 .6566982 _______________________ middle point:int exp( 20 ) 484165211 to 486165161 prime count: 99984 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 6502 6563 11722 5274 6831 8666 4869 3653 6652 3659 prime gap rates: for exp( 20 )-+ 99984 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.300608 1.31281 2.344775 1.054969 1.366419 1.733477 .9739558 .7307169 1.330613 .7319171 _______________________ middle point:int exp( 21 ) 1317765769 to 1319865707 prime count: 99943 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 6237 6305 11152 5139 6683 8412 4644 3551 6488 3546 prime gap rates: for exp( 21 )-+ 99943 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.310517 1.324805 2.343256 1.079805 1.40423 1.767527 .9757962 .7461353 1.363257 .7450847 _______________________ middle point:int exp( 22 ) 3583812847 to 3586012819 prime count: 99865 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 6003 5996 10815 4749 6279 8331 4559 3434 6378 3647 prime gap rates: for exp( 22 )-+ 99865 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.322445 1.320903 2.382516 1.046192 1.383247 1.835298 1.004336 .7565013 1.405057 .8034246 _______________________ middle point:int exp( 23 ) 9743653463 to 9745953401 prime count: 100034 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 5736 5684 10445 4750 5975 8191 4474 3463 6332 3554 prime gap rates: for exp( 23 )-+ 100034 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.318832 1.306876 2.401533 1.092129 1.373783 1.88329 1.02867 .7962193 1.455865 .8171422 _______________________ middle point:int exp( 24 ) 26487922141 to 26490322127 prime count: 100279 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 5605 5479 9897 4483 5873 7828 4466 3454 6266 3635 prime gap rates: for exp( 24 )-+ 100279 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.341457 1.311301 2.368671 1.072927 1.405598 1.873493 1.068858 .8266537 1.499656 .8699728 _______________________ middle point:int exp( 25 ) 72003649387 to 72006149329 prime count: 99749 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 5226 5251 9754 4290 5733 7575 4360 3289 6045 3505 prime gap rates: for exp( 25 )-+ 99749 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.309788 1.316053 2.444636 1.075199 1.436857 1.898515 1.092743 .8243191 1.515053 .8784549 _______________________ middle point:int exp( 26 ) 195728309461 to 195730909427 prime count: 100075 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 5149 5089 9344 4221 5551 7339 4169 3307 6060 3490 prime gap rates: for exp( 26 )-+ 100075 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.337737 1.322148 2.427619 1.096637 1.442178 1.90671 1.083128 .8591756 1.574419 .90672 _______________________ middle point:int exp( 27 ) 532046890633 to 532049590591 prime count: 100228 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 5015 4880 9010 4090 5409 7168 4115 3271 5974 3366 prime gap rates: for exp( 27 )-+ 100228 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.35097 1.314603 2.427166 1.101788 1.457108 1.930957 1.108523 .881161 1.609311 .9067526 _______________________ middle point:int exp( 28 ) 1446255664351 to 1446258464263 prime count: 100125 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 4764 4770 8613 4069 5163 6968 4040 3151 5820 3360 prime gap rates: for exp( 28 )-+ 100125 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.332255 1.333933 2.408629 1.137898 1.443835 1.948604 1.129788 .8811785 1.627566 .9396254 _______________________ middle point:int exp( 29 ) 3931332847177 to 3931335747103 prime count: 99810 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 4461 4471 8341 3903 5037 7039 3864 3100 5716 3433 prime gap rates: for exp( 29 )-+ 99810 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.296153 1.299058 2.423495 1.134025 1.463511 2.045196 1.122693 .9007114 1.660796 .9974652 _______________________ middle point:int exp( 30 ) 10686473081527 to 10686476081483 prime count: 99765 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 4418 4361 8076 3766 5005 6699 3780 3030 5602 3299 prime gap rates: for exp( 30 )-+ 99765 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.328522 1.311382 2.428507 1.132461 1.505037 2.014434 1.136671 .9111412 1.684559 .9920313 _______________________ middle point:int exp( 31 ) 29048848115317 to 29048851215247 prime count: 99631 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 4269 4281 7920 3601 4761 6462 3734 2905 5536 3171 prime gap rates: for exp( 31 )-+ 99631 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.328291 1.332025 2.464293 1.120444 1.481376 2.010639 1.161827 .9038854 1.722516 .9866508 _______________________ middle point:int exp( 32 ) 78962958582769 to 78962961782659 prime count: 100059 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 4180 4209 7657 3671 4739 6329 3667 2905 5369 3322 prime gap rates: for exp( 32 )-+ 100059 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.336811 1.346086 2.448795 1.174027 1.515586 2.024086 1.172748 .9290519 1.717067 1.062413 _______________________ middle point:int exp( 33 ) 214643578135919 to 214643581435909 prime count: 99803 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 4007 3997 7483 3410 4529 6331 3672 2768 5371 3088 prime gap rates: for exp( 33 )-+ 99803 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.32492 1.321614 2.474264 1.127521 1.49752 2.093354 1.214152 .915243 1.775929 1.021052 _______________________ middle point:int exp( 34 ) 583461740827463 to 583461744227449 prime count: 100008 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 3846 3827 7240 3397 4414 6178 3390 2742 5295 3171 prime gap rates: for exp( 34 )-+ 100008 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.307535 1.301076 2.461403 1.154888 1.50064 2.100352 1.152508 .9322054 1.800156 1.078054 _______________________ middle point:int exp( 35 ) 1586013450563471 to 1586013454063421 prime count: 100221 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 3790 3719 7090 3339 4477 6109 3430 2752 5256 3105 prime gap rates: for exp( 35 )-+ 100221 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.323575 1.29878 2.476028 1.166073 1.563495 2.133435 1.197853 .961076 1.835543 1.084354 _______________________ middle point:int exp( 36 ) 4311231545315237 to 4311231548915191 prime count: 100288 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 3712 3659 6808 3304 4261 5808 3422 2706 5021 3108 prime gap rates: for exp( 36 )-+ 100288 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.332482 1.313457 2.443842 1.186024 1.529555 2.084876 1.228382 .9713625 1.802369 1.115667 _______________________ middle point:int exp( 37 ) 11719142370952669 to 11719142374652593 prime count: 99997 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 3507 3656 6643 3181 4074 5767 3381 2585 5018 3150 prime gap rates: for exp( 37 )-+ 99997 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.297629 1.352761 2.457984 1.177005 1.507425 2.133854 1.251008 .9564787 1.856716 1.165535 _______________________ middle point:int exp( 38 ) 31855931755213757 to 31855931759013613 prime count: 99858 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 3418 3479 6660 3040 4084 5581 3287 2518 4871 2975 prime gap rates: for exp( 38 )-+ 99858 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.300687 1.3239 2.534399 1.156843 1.554127 2.123796 1.250836 .9582006 1.853612 1.132108 _______________________ middle point:int exp( 39 ) 86593400422043939 to 86593400425943737 prime count: 100169 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 3474 3383 6441 2987 4014 5413 3173 2550 4916 2930 prime gap rates: for exp( 39 )-+ 100169 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.352574 1.317144 2.507752 1.162965 1.562819 2.107508 1.235382 .9928221 1.914005 1.140772 _______________________ middle point:int exp( 40 ) 235385266835020007 to 235385266839019877 prime count: 100029 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 3260 3299 6192 2901 3944 5416 3127 2584 4794 2840 prime gap rates: for exp( 40 )-+ 100029 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.303622 1.319217 2.476082 1.160064 1.577143 2.165772 1.250437 1.0333 1.917044 1.135671 _______________________ middle point:int exp( 41 ) 639843493528004981 to 639843493532104937 prime count: 99900 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 3272 3315 6144 2918 3708 5210 3074 2382 4695 2835 prime gap rates: for exp( 41 )-+ 99900 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.342863 1.36051 2.521562 1.197578 1.521802 2.138238 1.261602 .9775976 1.926877 1.163514 _______________________ middle point:int exp( 42 ) 1739274941518401059 to 1739274941522601019 prime count: 100205 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 3148 3147 5932 2732 3764 5302 3013 2310 4595 2850 prime gap rates: for exp( 42 )-+ 100205 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.319455 1.319036 2.486343 1.145093 1.577646 2.222284 1.262871 .9682152 1.925952 1.194551 _______________________ middle point:int exp( 43 ) 4727839468227196567 to 4727839468231496509 prime count: 100160 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 3159 3036 5859 2796 3722 5011 2972 2367 4613 2793 prime gap rates: for exp( 43 )-+ 100160 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.3562 1.303395 2.515345 1.200359 1.597903 2.151288 1.275918 1.016184 1.980421 1.199072 _______________________ middle point:int exp( 44 ) 12851600114357108327 to 12851600114361508271 prime count: 100104 g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 3049 3036 5699 2621 3599 5082 2975 2313 4510 2665 prime gap rates: for exp( 44 )-+ 100104 primes g 2 g 4 g 6 g 8 g 10 g 12 g 14 g 16 g 18 g 20 1.340166 1.334452 2.504955 1.152042 1.581915 2.233757 1.30764 1.016663 1.982338 1.171382 _______________________
 2018-08-18, 13:44 #3 hal1se   Jul 2018 2716 Posts equality surprise if number >210 then 7#, prime template: (2-1)(3-1)(5-1)(7-1)=48, so only 48 combination. 210N+{1,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,121,127,131,137,139,143,149,151, __________t__c__t__c_____t_____c__t__c________t_____c__t_____c________c___t___c___t___c___________c_______t___c_______t____ 157,163,167,169,173,179,181,187,191,193,197,199,209},{+211,... _______c___t___c_______t_______c___t___c___t________t t:probably twin (g2) c:probably cousin (g4) 15 probably twin = 15 probably cousin my hands and eyes does mistake sometimes, please forgive! so, i tested by inteligent calculator. ======================= primoryel-7- ____________________________________________________________ 1 209 48 ____________________________________________________________ probably g2 15 probably g4 15 ======================= primoryel-11- ____________________________________________________________ 1 2309 480 ____________________________________________________________ probably g2 135 probably g4 135 ======================= primoryel-13- 30030N+{1,....,30029} 5760 elements, in the prime template. ____________________________________________________________ 1 30029 5760 elements ____________________________________________________________ probabbly g2 1485 probabbly g4 1485 ======================= primoryel-17- ____________________________________________________________ 1 510509 92160 ____________________________________________________________ probably g2 22275 probably g4 22275 ======================= primoryel-19- ____________________________________________________________ 1 9699689 1658880 elements ____________________________________________________________ probably g2 378675 probably g4 378675 ======================= primoryel-23- ____________________________________________________________ 1 223092869 36495360 elements ____________________________________________________________ probably g2 7952175 probably g4 7952175 ======================= primoryel-29- calculating this times! %60 ======================= primoryel-31- calculating ... need very big memory a few terabits RAM, so i think and find another quick method,only one, simple calculator: cheap mini note-book or tablet ======================= question: every prime template, do see we equality allways? probably g2 =? probably g4 any math prof? 5# to 23# template b1 archive need, please: b1.org https://www.dosyaupload.com/e2bO
 2018-08-19, 14:44 #4 hal1se   Jul 2018 478 Posts 3 question for beatifull prime template equality prime templates, a beatifull equality: if real integer number >2 then, every prime:2n+{1} (2-1)=1, only 1 combination:{1} _________ if number >6 then every prime numbers:6n+{1,5} (2-1)(3-1)=2, only 2 combination:{1,5} _________ if number >30 then every prime numbers:30n+{1,7,11,13,17,19,23,29} (2-1)(3-1)(5-1)=8, only 8 combination:{1,...,29} this 8 element modulo previous template: 1 mod 6=1 7 mod 6=1 11 mod 6=5 17 mod 6=5 19 mod 6=1 23 mod 6=5 29 mod 6=1 {5# template elements } mod 6: 1: 4 times 5: 4 times 4=4 equal! __________ if number >210 then every prime numbers:210n+{1,11,...,209} (2-1)(3-1)(5-1)(7-1)=48, only 48 combination:{1,...,209} this 8 element modulo previous template: {7# template elements } mod 6: 1:24 times 5:24 times 24=24 equal! {7# template 48 elements } mod 30: 1:6 times 7:6 times 11:6 times ... 29:6 times 6=6=...=6 equal! _________ template 11#,...,17# equality ok, all perivous template modulos. very beatifull equaltiy! _________ template 19# if number >2*3*5*7*11*13*17*19 then: every prime numbers: 9699690n+{1,23,....,9699689} (2-1)(3-1)(5-1)(7-1)(11-1)(13-1)(17-1)(19-1)=1658880, only 1658880 combination! this 1658880 elements modulos 6,30,210,...,510510 modulo 6: 1:829440 times 5:829440 times 829440=829440 equal! modulo 30: 1:207360 times 7:207360 times ... 29:207360 times 207360 =207360 =...=207360 equal! modulo 210: 1:34560 11:34560 ... 209:34560 34560=34560=...=34560 equal! modulo 11# to 13# equality ok! modulo 17# 1: 18 19: 18 23: 18 29: 18 31: 18 .... 510479: 18 510481: 18 510487: 18 510491: 18 510509: 18 18=18=...=18 equality! ___________ template 23# if number >2*3*5*7*11*13*17*19*23 then: every prime numbers: 223092870n+{1,29,....,223092869} (2-1)(3-1)(5-1)(7-1)(11-1)(13-1)(17-1)(19-1)(23-1)=36495360, only 36495360 combination! inside 36495360 elements: {1,...,223092869} this 36495360 elements modulos 6,30,210,...,510510,9699690 modulo 30: 1 4561920 7 4561920 11 4561920 13 4561920 17 4561920 19 4561920 23 4561920 29 4561920 4561920 equal 8 times! this 36495360 elements modulo 210: 1 760320 times 11 760320 13 760320 17 760320 19 760320 23 760320 29 760320 31 760320 37 760320 41 760320 43 760320 47 760320 53 760320 59 760320 61 760320 67 760320 71 760320 73 760320 79 760320 83 760320 89 760320 97 760320 101 760320 103 760320 107 760320 109 760320 113 760320 121 760320 127 760320 131 760320 137 760320 139 760320 143 760320 149 760320 151 760320 157 760320 163 760320 167 760320 169 760320 173 760320 179 760320 181 760320 187 760320 191 760320 193 760320 197 760320 199 760320 209 760320 48 times 760320=760320=760320=...=760320 modulo 11# to 17# equaltiy ok. please test yoruself! modulo 19#: 1 22 23 22 29 22 31 22 37 22 41 22 ... 969949 22 969953 22 969959 22 969961 22 969967 22 22=22=...=22 165888 times =22 question 1: every prime template this equality posible allways? any math prof? _____________ small prime gap equality: p1210 then 7#, prime template: (2-1)(3-1)(5-1)(7-1)=48, so only 48 combination. 210N+{1,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,121,127,131,137,139,143,149,151, __________t__c__t__c_____t_____c__t__c________t_____c__t_____c________c___t___c___t___c___________c_______t___c_______t____ 157,163,167,169,173,179,181,187,191,193,197,199,209},{+211,... _______c___t___c_______t_______c___t___c___t________t t:probably twin (g2) c:probably cousin (g4) 15 probably twin = 15 probably cousin my hands and eyes does mistake sometimes, please forgive! so, i tested by inteligent calculator. ======================= primoryel-7- ____________________________________________________________ 1 209 48 ____________________________________________________________ probably g2 15 probably g4 15 ======================= primoryel-11- ____________________________________________________________ 1 2309 480 ____________________________________________________________ probably g2 135 probably g4 135 ======================= primoryel-13- 30030N+{1,....,30029} 5760 elements, in the prime template. ____________________________________________________________ 1 30029 5760 elements ____________________________________________________________ probabbly g2 1485 probabbly g4 1485 ======================= primoryel-17- ____________________________________________________________ 1 510509 92160 ____________________________________________________________ probably g2 22275 probably g4 22275 ======================= primoryel-19- ____________________________________________________________ 1 9699689 1658880 elements ____________________________________________________________ probably g2 378675 probably g4 378675 ======================= primoryel-23- ____________________________________________________________ 1 223092869 36495360 elements ____________________________________________________________ probably g2 7952175 probably g4 7952175 ======================= ======================= question 2: every prime template, do see we equality allways? probably g2 =? probably g4 any math prof? i calculated 5# to 23# template : please download: https://www.dosyaupload.com/e2bO _____________ sextuplet equality: sextuplet: 6 primes, p, p+4, p+6, p+10, p+12, p+16 another mean: 6 primes: 30n+7, 30n+11, 30n+13, 30n+17, 30n+19, 30n+23 if number>96 then eery sextuplet first number p: (p-97) mod 210 = 0 allways! so: ((p-97)/210) mod 11 only 5 combination: 0,4,5,6,10 another mean: p mod 2310 simple code! ((p-97)/210) mod (11*13) only 35 combination. another mean: p mod 30030 simple code! ((p-97)/210) mod (11*13*17) only 385 combination. ((p-97)/210) mod (11*13*17*19) only 5005 combination. ((p-97)/210) mod (11*13*17*19*23) only 85085 combination. any mean 5,35,385,5005,85085,... 5,5*7,5*7*11,5*7*11*13,5*7*11*13*17,5*7*11*13*17*19,... question 3: this regularly posible every sextuplet? note:sextuplet first number:p < trillion^(trillion^trillion) then ok! but this is not math prof! any math prof?
 2018-08-19, 14:55 #5 hal1se   Jul 2018 3·13 Posts sory small prime gap equality: p1
2018-08-19, 15:29   #6
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

22×2,281 Posts

Quote:
 Originally Posted by hal1se small prime gap equality: p1
This is profound!

 2018-08-19, 19:43 #7 hal1se   Jul 2018 3×13 Posts sory2 (23# template 1658880 elements) modulo (19#=9699690) 1 22 times 23 22 29 22 31 22 37 22 41 22 43 22 .... 9699643 22 9699647 22 9699649 22 9699653 22 9699659 22 9699661 22 9699667 22 9699689 22 1658880 elements, 22 = 22 = ... = 22: 1658880 times =22 why? important question? how?
 2018-08-19, 22:04 #8 hal1se   Jul 2018 3×13 Posts how is it? 19# to 23# real prime count=11637502 middle point of range: 19# *((23+1)/2)= 19# * 12=116396280 ln 116396280=18,57251113398803281112229198418 twin or cousin prime count lower limit= int[(4/3)*range prime count/ln(middle point of range)]= int[(4/3)*11637502/(ln 116396280)]=835464 real twin prime count=838609 real cousin prime count=839672 abs(deviation) < %0,5 deiation negative! lower limit! _______ 23# to 29# real prime count=288086265 middle point of range: 23# *((29+1)/2)= 23# * 15=3346393050 ln 3346393050=21,9311489012313922576953399063 twin or cousin prime count lower limit= int[(4/3)*range prime count/ln(middle point of range)]= int[(4/3)*288086265 /(ln 3346393050)]=17514587 real twin prime count=17567651 real cousin prime count=17563582 abs(deviation) < %0,3 deviation negative! lower limit! ________ 37# to 41# real prime count=9155428058351 middle point of range: 37# *((41+1)/2)= 37# * 21=155835500831010 ln 155835500831010=32,679822084968449905680465344453 twin or cousin prime count lower limit= int[(4/3)*range prime count/ln(middle point of range)]= int[(4/3)*9155428058351 /(ln 155835500831010)]=373540510085 real twin prime count=please count yourself real cousin prime count=please count yourself abs(deviation) < % 1 _________ 47# to 53# real prime count=726840450530910033 middle point of range: 47# *((53+1)/2)= 47# * 27=16602024129889268070 ln 16602024129889268070=44,25605629735728479752615963101 twin or cousin prime count lower limit= int[(4/3)*range prime count/ln(middle point of range)]= int[(4/3)*726840450530910033 /(ln 16602024129889268070)] =21898033439679764 real twin prime count=please count yourself real cousin prime count=please count yourself abs(deviation) < % 1 note 53#>2^64 if are you number lover: not problem this test! if are you real math man or math girl: don't test. only think how is it?
2018-08-21, 10:48   #9
hal1se

Jul 2018

3×13 Posts
very simple! if think it and look full picture!

p1<p2<p3
gap:p2-p1
another gap: p3-p2
but not: p3-p1

Quote:
 Originally Posted by Batalov This is profound!
[sequantial primes:p1,p2,p3]
p1<p2<p3
1)p2-p1: a prime gap
2)p3-p2: another gap
3)"but not: p3-p1" , contain [... bla bla]
so [...] not need!

if middle point of range > exp(13) then:
and
if
(ln(middle point of range))/ln(range)<exp(1) then

lower limit formula:

(16/{16,12,4,1})*range/(ln(middle-point-of-range)^{1,2,4,6})

another lower limit formula:
(16/{12,4,1})*(real-prime-count-of-range)/(ln(middle-point-of-range)^{1,3,5})

note: this formula math prof, very simple!
if think it and look full picture!
but math people very slow!

2018-08-21, 13:15   #10
CRGreathouse

Aug 2006

2·5·593 Posts

Quote:
 Originally Posted by hal1se if middle point of range > exp(13)
I'm guessing this means that p2 >= 442439 and so p1 >= 442399?

Quote:
 Originally Posted by hal1se and if (ln(middle point of range))/ln(range)
This part is inscrutable to me, even after studying your other posts. What is the logarithm of a range of numbers?

Quote:
 Originally Posted by hal1se then lower limit formula: (16/{16,12,4,1})*range/(ln(middle-point-of-range)^{1,2,4,6}) {prime,twin,quadruplet,sextuplet}
Now you're using array-valued constants without any indication of how you want us to handle them. Your dimensions don't match up at all.

You're going to have to learn some communication skills (and some math) before this is going to make any progress.

2018-08-25, 12:42   #11
hal1se

Jul 2018

3×13 Posts
good by number lovers

Quote:
 Originally Posted by CRGreathouse I'm guessing this means that p2 >= 442439 and so p1 >= 442399? This part is inscrutable to me, even after studying your other posts. What is the logarithm of a range of numbers? Now you're using array-valued constants without any indication of how you want us to handle them. Your dimensions don't match up at all. You're going to have to learn some communication skills (and some math) before this is going to make any progress.

if middle point of range > exp(13)
I'm guessing this means that p2 >= 442439 and so p1 >= 442399?

wrong mean!

if middle point of range > exp(13) : this mean only for: very large range prime or k-tuplets count. not any mean: p1,p2

What is the logarithm of a range of numbers?

if range mini mini than, not lower limit! this criteria for large ranges.

Now you're using array-valued constants without any indication of how you want us to handle them. Your dimensions don't match up at all.

primes and k-tuplets very regularly or rough regularly, but regularly.
my calculation very rough, but meanfull!

for example: septuplet or 7tuplets:
oeis.org/A022009
p, p + 2, p + 6, p + 8, p + 12, p + 18, p + 20

another mean: 30n+11, 30n+13, 30n+17, 30n+19, 30n+23, 30n+29, 30n+30+1

11, 165701, 1068701, 11900501, 15760091, 18504371, 21036131, 25658441, 39431921, 45002591, 67816361, 86818211, 93625991, 124716071, 136261241, 140117051, 154635191, 162189101, 182403491, 186484211, 187029371, 190514321, 198453371, ...

rough formula (oeis.org/A022009)) septuplet:
if middle point of range > exp(30) [because number > exp(30) septuplets are small fluctations, but number < exp(30) big fluctations]
if ln(middle point of range)/ln(range)>exp(1) than [because need very large ranges]

rough septuplet count formula=16/{2/7}*range/(ln(middle point of range))^7

this rough calculation elements: 16/{2/7}=56 only handly and not important!

important subject:(ln(middle point of range))^7

____________
range: int[exp(43)]-2e12 to int[exp(43)]+2e12
4727837468229346561 to 4727841468229346561
range:4 trillion, 19 decimal digit, real positive integer.

ln(middle point of range)=43
7tuplet rough range count=?56*4*1e12/43^7=824
(oeis.org/A022009)) septuplet mean: 30n+11, 30n+13, 30n+17, 30n+19, 30n+23, 30n+29, 30n+30+1
_____________

range: int[exp(41))-2e12 to int[exp(41)]+2e12
range: 639841493530054949 to 639845493530054949
range:4 trillion, 18 decimal digit, real positive integer.

ln(middle point of range)=41
7tuplet rough range count=? 56*4*1e12/41^7=1150
______________

range: int[exp(39)]-2e12 to int[exp(39)]+2e12
range: 86591400423993746 to 86595400423993746
range:4 trillion, 17 decimal digit, real positive integer.

ln(middle point of range)=39
7tuplet rough range count=? 56*4*1e12/39^7=408*4=1632
____________

range: int[exp(36)]-2e12 to int[exp(36)]+2e12
range: 4309231547115195 to 4313231547115195
range:4 trillion, 16 decimal digit, real positive integer.

ln(middle point of range)=36
7tuplet rough range count=? 56*4*1e12/36^7=2858
____________

range: int[exp(33)]-2e12 to int[exp(33)]+2e12
range: 212643579785916 to 216643579785916
range:4 trillion, 15 decimal digit, real positive integer.

ln(middle point of range)=33
7tuplet rough range count=? 56*4*1e12/33^7=5255

19,18,17,16,15 decimal digit, range: 4 trillion integers, fix!
7tuplet lower limit formula's part: 56*4*1e12 fix!
only 43^7,41^7,39^7,36^7,33^7 are different!
_____________

i am only number lover!
i love number test for regular basis look!
i am an autistic, alzheimer, parcinson, etc! brain damage!
i can not many learn.
my brain damage but spesific.
i look full picture sometimes!

please test and please think: how is it, at least rough regularly septuplet or 7tuplet!
important question: how is it?
if someone chooses, septuplets mean : (oeis.org/A022009) and (oeis.org/A022010) combination same time then, not regularly!
but septuplets mean : only (oeis.org/A022009) , at least rough regularly!
because every prime template's only direct element groups very regularly or at least rough regularly!
how is it, prime template direct elements regularly?
this is important question!
math people very slow for this answer!
may be, prime template elements equality, gives you important tips.
3 question for beatifull prime template equality, top lines!

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