quadruplets (so:twin primes) analysis every exponantial range

[hal1se]

quadruplets analysis:
__________________________________
exp(-+0,5+21)

799902177
2174359553

Prime numbers: 65206481
Prime quadruplets: 28890

Elapsed time: 0.40 sec

int(4*exp (21) / 21^4 )= 27124 < 28890
deviation:(27127-28890)/27124= % -6,5 negative: lower limit count!
__________________________________
exp(-+0,5+22)

2174359553
5910522063

Prime numbers: 169221623
Prime quadruplets: 65506

Elapsed time: 1.30 sec

int(exp(22) / 22^4 )= 61213 < 65506
deviation: (61213 - 65506) / 61213 = % -7,0
___________________________________
exp(-+0,5+23)


5910522063
16066464720

Prime numbers: 440059184
Twin primes: 25182309
Prime triplets: 4723257
Prime quadruplets: 148557
Prime quintuplets: 31336
Prime sextuplets: 1171

Elapsed time: 3.82 sec

int(4*exp(23)/(23^4)) = 139290 < 148557
deviation: (139290 - 148557) / 139290 = % -6,7
___________________________________
exp(-+0,5+24)

16066464720
43673179097

Prime numbers: 1146515015
Prime quadruplets: 341304

Elapsed time: 11.68 sec

int(4*exp(24)/(24^4)) = 319361 < 341304
deviation: (319361 - 341304) / 319361 = % -6,9
___________________________________
exp(-+0,5 + 25)

43673179097
118716009132
Prime numbers: 2992276391
Prime quadruplets: 787283

Elapsed time: 40.63 sec



int(4*exp(25)/(25^4)) = 737330 < 787283
deviation: (737330 - 787283) / 737330 = % -6,8
___________________________________
exp(-+0,5 + 26)


118716009132
322703570371

Prime numbers: 7821928491
Prime quadruplets: 1834796

Elapsed time: 126.90 sec



int(4*exp(26)/(26^4)) = 1713259 < 1834796
deviation: (1713259 - 1834796) / 1713259 = % -7,09
___________________________________
exp(-+0,5 + 27)


322703570371
877199251318

Prime numbers: 20476919479
Prime quadruplets: 4284690

Elapsed time: 372.68 sec



int(4*exp(27)/(27^4)) = 4004570 < 4284690
deviation: (4004570 - 4284690) / 4004570 = % -6,9950
___________________________________
exp(-+0,5 + 28)

877199251318
2384474784797
Prime numbers: 53679503762
Prime quadruplets: 10069640

Elapsed time: 1106.61 sec


int(4*exp(28)/(28^4)) = 9411814 < 10069640
deviation: (9411814 - 10069640) / 9411814 = % -6,989
___________________________________
exp(-+0,5 + 29)

2384474784797
6481674477934
Prime numbers: 140897751078
Prime quadruplets: 23801213
Elapsed time: 3879.80 sec
_________________________________
0 to exp(31+0,5)
0
47893456332462
0 to exp (31,5)
Prime numbers: 1572095341867
Prime quadruplets: 232653982
_________________________________
exp(32-+0,5)
exp(31,5) to exp(32,5)
Prime numbers: 2565328558735
Prime quadruplets: 322735035

0 to exp(31,5) prime count < exp(32-+0,5) prime count!
0 to exp(31,5) quadruplets count < exp(32-+0,5) quadruplets count!
_________________________________
exp(33-+0,5)
130187912050633
353887435612260

Prime numbers: 6762467049487
Prime quadruplets: 775878111

__________________________
exp(34-+0,5)
353887435612260
961965785544776

Prime numbers: 17842861844016
Prime quadruplets: 1872127524
________________________________

0 to exp(33,5) quadruplets count (232653982+322735035+775878111=1331267128)< exp(34-+0,5) quadruplets count:1872127524
(every range quadruplets count ) > (2 to previous range cumulative count!)
_________________________________
rough lower limit count quadruplets for range:exp(34-+0,5)

int(4*exp (34) / (34^4) )= 1746452217 < 1872127524
deviation:(1746452217-1872127524)/1746452217= % -7,1 negative: lower limit count!

_________
exp(37,777) -+ 500e9:
1 trillion 16 digit integer.
25487904036980675
25488904036980675

Prime numbers:26471172237
Prime quadruplets:2039480

Elapsed time:956.16 sec
16 minutes

rough calc:int[4*(10^12)/((37,777)^4)]=1964039 < 2039480 :real count
____________
exp(44-+0.5)
one computer time!>100 years
but parellel more than 1e6 computer < 1 day
and out of range exp(44,5) >2^64, but not problem, easily ok, if 128 bit computer near future.
exp(44)-+(500e9):
range:one trillion 20 digit integer.
12851599614359308275
12851600614359308275
Prime numbers:22727213027
Prime quadruplets:1105744

Elapsed time:8622.31 sec
2,5 hours

rough calculation:
range:10^12
logarithym natural of middle point of range=44

rough calculation:int(4*10^12/(44^4))=1067208 < real count Prime quadruplets:1105744
rough lower limit count!
exp(N-+0,5) middle point, is not exp(N) of course! but near, and this calculation is very rough!
_______

question: is it a regularity every range, N>13 and N is an integer or not integer,
if range:exp(N-+0,5): real count > rough lower limit count?
if is it every range true?
answers: yes! than: infinity quadruplets are there!
every quadruplets contain two twin prime: and so: infinty twin prime are there.

any proof math?

[hal1se]

N>13, N is an integer or not!
every exp(N - + 0,5) range rough lower limit sextuplet prime count:
int[16*(exp(N))/(N^6)]

here:
int[...]: for example: int[2,718]=2
^: for example 10^4=10000

sextuplet math mean?
if sextuplet first number>96 then
every sextuplet first number p:
(p-97) mod 210 = 0. allways!
((p-97)/210) mod 11 only 5 combination: 0,4,5,6,10, another mean: p mod 2310, simple code!
((p-97)/210) mod (11x13) only 35 combination. (another mean p mod 30030, simple code!)
((p-97)/210) mod (11x13x17) only 385 combination.
((p-97)/210) mod (11x13x17x19) only 5005 combination.
((p-97)/210) mod (11x13x17x19x23) only 85085 combination.
....
any mean? 5,35,385,5005,85085,...?
5,5x7,5x7x11,x5x7x11x13,5x7x11x13x17,...?
a beatifull regularity.
this regularity only for sextuplet and contain pi=~3,14159653{5,6}