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Old 2018-09-02, 16:50   #1
hal1se
 
Jul 2018

478 Posts
Default every exponantial ranges brun's sum: very slow go to low!


http://www.britannica.com/science/twin-prime-conjecture
In 2010 Nicely gave a value for Brun’s constant of
1.902160583209 ± 0.000000000781 based on all twin primes less than 2 × 10^16
0.000000000781
__123456789012
1,902160583209+0,000000000781=
1,90216058399
1,902160583209-0,000000000781=
1,902160582428 **
__12345678
8 digit after comma!
range 2e16 to 1e17
real prime count:2075693725704225
ln(middle point of range)=
=ln((2e16+1e17)/2)=ln(6e16)=38,633120957132785945100340633331
rough twin count=
=1,3219*(2075693725704225/38,633120957132785945100340633331=
=71023501804397 -+(%1)

range first twin:
(20000000000000129, 20000000000000131)
range last twin:
(99999999999998807, 99999999999998809)
99999999999998807=~1e17
99999999999998807 so 17 decimal digits, difference microscopic! for 1e17=10^18, so 18 decimal digits

range_brun_sum > range_twin_count*(2*1/1e17)
0,001420=range_twin_count*(2*1/1e17)

range_brun_sum=1/20000000000000129 + 1/20000000000000131 + ... + 1/99999999999998807 + 1/99999999999998809


range_brun_sum [newer less] range_twin_count*(2* 1/1e17)

71023501804397*2/1e17=0,001420
0,001420
__123
3th digit after comma :1
1,902160582428 **
__12345678
4. digit after comma not meanfull!
so:
1,902 only true!
if some one say, your calc. rough!
not important rough!
we use last range element:2*1/1e17
range_brun_sum > range_twin_count*(2*1/1e17)
so, may be:0,001420 up to another great number!
but 2. digit not wrong.(2. digit true only range 0 to 1e18)
only 4. and may be 3. digit after comma false!

range 1e17 to 1e18
range real prime count=22116397130086627
middle point of range=5,5e17
range rough twin count=1,3219*22116397130086627/ln(5,5e17)=715706232480603 -+(%1)
715706232480603*(2/ 1e18)=0,001431
if range rough twin count <%1 real twin count:
715706232480603*99/100=708549170155796
708549170155796*(2 /1e18)=0,001417
0,001431=~0,001417 near!
not important %1 rough twin count!

range 1e17 to 1e18:0,0014... same before 2e16 to 1e17 range 0,0014...
this 0,0014s must be large number, so:this two range_brun_sums must be >0,0014..

range 1e18 to 1e19
range real prime count:209317712988603747
middle point of range=5,5e18
range rough twin count=1,3219*209317712988603747/ln(5,5e18)=6412256746480331 -+(%1)
6412256746480331*(2/ 1e19)=0,001282...

stop!
zem!

this 3 range bruns' sum 0,001282+0,001417+0,00142=0,005539
0,005539
__123
after comma 3. digit: 5
0 to 2e16 sieve brun sum=1.90216...
1,90216058+0,005539=1,90769958

http://www.britannica.com/science/twin-prime-conjecture
In 2010 Nicely gave a value for Brun’s constant of
1.902160583209 ± 0.000000000781 based on all twin primes less than 2 × 10^16
0.000000000781
__123456789012

12 or 8 digit only ilizione!

dear Nicely! only 2 digit after coma true! 1,90
brun sum must be > 1,907
very clear!

analysis brun sum:
i am an autistic, brain damage!
last day same one say:
twin-prime-conjecture
so i search at duckduckgo
i saw brun's sum.
this time analysis it:
range: exp(n) to exp(n+1)
for example:
____
23
24
4,783989444169075e-03
1,79213361739538

this mean:
range: int(exp(23)) to int(exp(24))
4,783989444169075e-03 : this mean :range brun sum
1,79213361739538 : this mean 0 to range upper bound cumulative brun sum

please forgive this paste, but please look every one this results:
1,5 hours test: 18 floating point variables but only 15 digit print out!
____
0
1
0
0
____
1
2
,8761904761904762
,8761904761904762
____
2
3
,279287276191301
1,155477752381777
____
3
4
,1143868809927461
1,269864633374523
____
4
5
,1137490813040892
1,383613714678613
____
5
6
8,872646653751062e-02
1,472340181216123
____
6
7
5,521056269861256e-02
1,527550743914736
____
7
8
,0440171676677505
1,571567911582486
____
8
9
3,913401191595095e-02
1,610701923498437
____
9
10
2,808981294480608e-02
1,638791736443243
____
10
11
2,353587031749571e-02
1,66232760676074
____
11
12
1,982513665126122e-02
1,682152743412
____
12
13
1,675856631072195e-02
1,698911309722722
____
13
14
1,449627888590353e-02
1,713407588608626
____
14
15
1,268681390107178e-02
1,726094402509697
____
15
16
1,104899602949351e-02
1,737143398539191
____
16
17
9,708477915365204e-03
1,746851876454556
____
17
18
8,609915072354517e-03
1,755461791526911
____
18
19
7,720212327956113e-03
1,763182003854867

____
19
20
6,946011873574503e-03
1,770128015728441
____
20
21
6,28638488254902e-03
1,77641440061099
____
21
22
5,716112332965187e-03
1,782130512943955
____
22
23
5,219115007254201e-03
1,78734962795121
____
23
24
4,783989444169075e-03
1,79213361739538


only look every range sum please.
for examle: 4,783989444169075e-03
how mean only this range sum?

every exponantial range brun's sums very slow go to low!

some one test million cores 128 bit computer, 38 floating point calculation, 0 to 1e29
for twins < 2^40 then clasic sieve and twins > 2^40 fast sieve, and even million cores and time > a few hundres years:
sum result only after comma only 5. digit meanfull!

may be: brun sum > 1,9, don't forget!




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Old 2018-09-03, 07:18   #2
hal1se
 
Jul 2018

478 Posts
Default every exponantial ranges, brun sum regularly!

question: average reciproc element=avr?

____
22
23
5,219115007254201e-03 :range brun sum
range exp22 to exp23: real twin count:15957923
2*avr*real twin count=range brun sum
avr=range brun sum/(2*real twin count)=
5,219115007254201e-03/(2*15957923)=1,6352739035193367582986833562237e-10
ln(reciproc(1,6352739035193367582986833562237e-10))=22,534040614533721383423945094182

____
23
24
4,783989444169075e-03
ln(reciproc(range brun sum/(2*real twin count)))=
=ln(1/(4,783989444169075e-03 /(2*39772849)))
=23,534322700912901121694938047962

____
21
22
5,716112332965187e-03
1,782130512943955
ln(1/(range brun sum/(2*real twin count)))=
=ln(1/(5,716112332965187e-03/ (2*6427922)))=21,533775418648340934467457654403
____
13
14
1,449627888590353e-02
ln(1/(range brun sum/(2*real twin count)))=
=ln(1/(1,449627888590353e-02 / (2*5472)))=13,534409930911060546669441893616

does see some one any mean?
exp 13 to exp 14
avr=13,5+0,034409930911060546669441893616

exp 23 to 24
avr=23,5+0,034322700912901121694938047962
avr remainder diffrence exp13 and exp 23:
0,0344099-0,0343227=0,0000872 very low difference!

exp 21 to 22
avr remainder:
0,03377541
avr remainder diffrence exp21 and exp 23:
0,0343227 - 0,03377541 = 0,00054729 low difference! but not very low!

if twins in exponantial range windows, we see surprise regularly.
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Old 2018-09-03, 07:20   #3
hal1se
 
Jul 2018

3·13 Posts
Default 10^n range test up to n=11, range brun sum very slow go to small!

question:
if range 10^n, brun sum:
this internet board many people, angry to me, for 'computer' print output, paste.
if Nicely, before test 2*10^16, only first a few 10^n range test, only a few hours!
He is see easly, this range brun sum very low go to small!

____
0
1
,8761904761904762
,8761904761904762
____
1
2
,4547998895286105
1,330990365719087
____
2
3
,1870420978405043
1,518032463559591
____
3
4
9,886109387260966e-02
1,616893557432201
____
4
5
5,590602739554085e-02
1,672799584827741
____
5
6
3,797734597647975e-02
1,710776930804221
____
6
7
2,758011311304998e-02
1,738357043917271
____
7
8
2,045857715070371e-02
1,758815621067975
____
8
9
1,592033657056097e-02
1,774735957638536
____
9
10
1,274254508070533e-02
1,787478502719241
____
10
11
1,042580823589026e-02
1,797904310955131


range 10^10 to 10^11
range brun sum>1e-02
1e-02=0,01

if, i will explain temporary result brun sum:
up to 10^11
1,797904310955131-+ bla bla
is this result ethic?

i must be say:
brun sum > 1,79 up to 10^11
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Old 2018-09-03, 09:36   #4
LaurV
Romulan Interpreter
 
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Jun 2011
Thailand

100011101111002 Posts
Default

continuing this crap, soon enough ye'll get your own blog and be restricted to blogorhea forum...
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Old 2018-09-05, 23:07   #5
hal1se
 
Jul 2018

3·13 Posts
Default how method brun sum calculate easy?

dear larry, please don't angry.
patient please!
if are you know 1e26 to 1e27 range brun sum lower limit, please calculate, and show me!
but not read before bottom efore calculation!

question: what is known, range brun sum lower limit?
for this answer, step by step:
question: big range twin count stabil?
range:exp(35-+0,5)
961965785544776
2614894114445696
Prime numbers: 47119213510151
Twin primes: 1773579320354
Elapsed time: 2330556.63 sec ~ 27 days (my old amd 4 cores 1,6Ghz 8GB Ram laptop)

ln(middle point of range): ln( (exp35,5 + exp34,5)/2 )=35,12011450695827752463176337351
ln: logarithm natural
please look at this result:
ts:twin stabil number: 1,32192 (only exp(n-+0,5), n>33)
1,32192*47119213510151 / 35,12011450695827752463176337351 = 1773565707224,4
1773565707224 : rough twin count.
deviation:(1773565707224-1773579320354)/1773579320354=-7,6e-6
replace 1,32192 to 4/3, so:if we take 4/3, this rational number, only 0 to exp(35,5) rough twin count. but this time we look only high stability!
range:exp(34-+0,5)
353887435612259
961965785544776
Prime numbers: 17842861844016
Twin primes: 691321034769

Elapsed time: 816749.49 sec ~ 9,5 days
ln(middle point of range): ln( (exp33,5 + exp34,5)/2 )=34,12011450695827752463176337351


1,32192*17842861844016/34,12011450695827752463176337351 = 691288299282,6
691288299282 : rough twin count.
deviation:(691288299282-691321034769)/691321034769=-4,7e-5

deviation negative: lower limit!
deviation very low!

range:
exp(-+0,5+33)
130187912050633
353887435612260

Prime numbers: 6762467049487
Twin primes: 269934079037


Elapsed time: 480536.52 sec ~ 5,5 days
ln(middle point of range):33,12011450695827752463176337351
0,1201145069582775246317633 : stabil!
1,32192*6762467049487/33,12011450695827752463176337351=269909708198,01
rough twin count=269909708198
deviation:(269909708198-269934079037)/269934079037=-9,028e-5
deviation: negative! this mean: lower limit!
deviation very low!: this mean previous big ranges very small fluctation, so very stabil twin count!

question: range 10^n windows twin count stabil?
test: range 1e10 to 1e11:
Primes: 3663002302
Twin primes: 196963369
Seconds: 24.877
1,32192*3663002302/ln(5,5e10)=195797764
deviation:(195797764-196963369)/196963369=-0,0059
deviation negative: lower limit!
abs(deviation)<0,006, so <%0,6
test2:range 1e11 to 1e12:
Primes: 33489857205
Twin primes: 1646209172
Seconds: 316.650
1,32192*33489857205/ln(5,5e11)=1637650668
deviation:(1637650668-1646209172)/1646209172=-0,0052
test3:range 1e12 to 1e13:




if range goes to big ranges then abs deviation goes to very low numer!
very important result!

question:what is known, prime counting 10^n range:
duckduckgo.com/?q=prime+counting+funtion
first result: https://en.wikipedia.org/wiki/Prime-counting_function
my zone bloc this internet type, but not important, i look tor browser easy:
range: ral prime count:
<10^26: 1699246750872437141327603
<10^27: 16352460426841680446427399
this is very important information for rough twin count:
1e26 to 1e27 range prime count:
16352460426841680446427399 - 1699246750872437141327603 = 14653213675969243305099796
rough twin count=1,32192*14653213675969243305099796/ln(5,5e26)=
=314597359935696445388506 -+%1 (abs(deviation) may be only <%0,1 but we must %1, if some one count real twin, everybody sure!)
=3,14597359935696445388506e+23
24 digit twin count!

question:
what is average reciproc element for 10^n ranges?
1e10 to 1e11 range:
____
10
11
1,042580823589026e-02 :range brun sum.
log(1/(range brun sum / (2* range twin count)))=
log( 1/(1,042580823589026e-02 / (2*196963369)) )=10,5773
important:log mean base:10 logarithm! not this log mean, ln:logarithm natural.
10,5 + 0,0773 : this mean 0,0773 average brun sum reciproc remainder!
if we take 2*0,0773=0,1546 lower limit range brun sum remainder.
if we take negative: -0,0773 remainder, this time upper limit range brun sum.

question: 1e26 to 1e27 range brun sum lower and upper limit=?
very easy:
range average reciproc remainder: 26,5 + 0,0773 = 26,5773

lower limit range reciproc element : 1/( 10^(26,5+2*0,0773) )=2,2151339841371935282722195619031e-27
lower limit brun sum=range twin count*(2* lower limit range reciproc element)=
=314597359935696445388506*(2* 2,2151339841371935282722195619031e-27)
=0,00139375

upper limit range reciproc element : 1/( 10^(26,5 -0,0773) )= 3,7783309865888608585373512944833e-27
upper limit brun sum=range twin count*(2* upper limit range reciproc element)=
=314597359935696445388506*(2*3,7783309865888608585373512944833e-27)=
=0,0023773

how mean this result?
range 1e26 to 1e27 only for this range brun sum > 0,00139375

simple say: range brun sum > 0,001
this mean: cumulative brun sum up to 1e27, after comma 3. digit not meanfull!



https://planetmath.org/BrunsConstant

B=∑pp+2⁢ is prime(1p+1p+2)≈1.9216058

after comma digit?

if we look exponantial range windows for brun sum,
exp(87) to exp(88) range brun sum lower limit > 1e-6
5. digit not meanfull!
exp(88)=1,6516e+38
up to 1e38, after comma 6. digit not stabil!

question: how calculate this 7.digit?
exp(88) to exp(89) rough twin count:
(4/3)*exp(88,5+0,12011450695827752463176337351)/((88,5+0,12011450695827752463176337351)^2)=
=5,2131574144461774557010443800843e+34
this twin count very rough!
twin count must be > 1e34
even we use combiatric method this 1e34 * 2 times element,
(1/(twin first prime) + 1/(twin first prime + 2) ) reciproc element sum how method?
2e34 sum only for this range: exp(88) to exp(89)
if we can, 1e18 reciproc sum per second by super computer,
2e34/1e18=2e16 seconds=~2e16/3600/24/365=634195839 years. but 6. digit not stabil, up to exp(88).

question again:
https://planetmath.org/BrunsConstant

B=∑pp+2⁢ is prime(1p+1p+2)≈1.9216058
how method after comma 7. digit?

question: this brun sum calculate 10. digit, but not use computer, only windows calculator, only < 1 minute!

answer: yes posible.

don't angry larry please!
only think larry please!
how method brun sum calculate easy?
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Old 2018-09-06, 04:17   #6
Batalov
 
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"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

33×73 Posts
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Welcome back, dear Raman!
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