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Old 2018-07-27, 10:27   #1
hal1se
 
Jul 2018

1001112 Posts
Default 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.
hal1se is offline   Reply With Quote
Old 2018-07-28, 17:50   #2
hal1se
 
Jul 2018

1001112 Posts
Default 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
_______________________


hal1se is offline   Reply With Quote
Old 2018-08-18, 13:44   #3
hal1se
 
Jul 2018

2716 Posts
Default 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

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Old 2018-08-19, 14:44   #4
hal1se
 
Jul 2018

478 Posts
Default 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:

p1<p2<p3
gap:p2-1
another gap: p3-p1
but not! p3-p1
small prime gap:
g2: p2-p1=2: twin primes
g4: p3-p2=4: cousin primes


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
=======================

=======================
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?





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Old 2018-08-19, 14:55   #5
hal1se
 
Jul 2018

3·13 Posts
Default sory

small prime gap equality:

p1<p2<p3
gap:p2-p1
another gap: p3-p2
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Old 2018-08-19, 15:29   #6
Batalov
 
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Mar 2008
Phi(4,2^7658614+1)/2

22×2,281 Posts
Default

Quote:
Originally Posted by hal1se View Post
small prime gap equality:

p1<p2<p3
gap:p2-p1
another gap: p3-p2
This is profound!
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Old 2018-08-19, 19:43   #7
hal1se
 
Jul 2018

3×13 Posts
Default 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?
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Old 2018-08-19, 22:04   #8
hal1se
 
Jul 2018

3×13 Posts
Default 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?
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Old 2018-08-21, 10:48   #9
hal1se
 
Jul 2018

3×13 Posts
Default 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 View Post
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})

{prime,twin,quadruplet,sextuplet}

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

{twin,quadruplet,sextuplet}


note: this formula math prof, very simple!
if think it and look full picture!
but math people very slow!
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Old 2018-08-21, 13:15   #10
CRGreathouse
 
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Aug 2006

2·5·593 Posts
Default

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

Quote:
Originally Posted by hal1se View Post
and
if
(ln(middle point of range))/ln(range)<exp(1)
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 View Post
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.
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Old 2018-08-25, 12:42   #11
hal1se
 
Jul 2018

3×13 Posts
Default good by number lovers

Quote:
Originally Posted by CRGreathouse View Post
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
7tuplet real count=please test yourself
_____________

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
7tuplet real count=please test yourself
______________

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
7tuplet real count=please test yourself
____________

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
7tuplet real count=please test yourself
____________


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
7tuplet real count=please test yourself

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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