﻿ Pair Prime number First P.I.CO file 　 Second 　 jsprime2.htm
Pair Prime number drawing board
Goldbach decomposition prime counting graph ; board 6 ; board 7
Goldbach conjecture. integer to prime summation
program 　 Arndt 　 jpg image list 　 Update 2017-03-29
XYGraph v2.3 - web page graph   ☜☞   donate   get code
Prime Counting Function Graph tute0058.htm#primeCountDraw click [A1]
Prime Counting >> Square Counting ; ∑(1/Sq.)=π*π/6 , ∑(1/prime)→∞
see proof at http://freeman2.com/tute0058.htm#docA403 to docA454
The Cauchy-Schwarz Master Class   J. Michael Steele   ★★★★★
Program environment Acer computer Aspire 5750Z-4830.
MSIE 9.0 Version:9.0.8112.16421
This file is personal home work. No one
If you suspect any view point wrong,
Freeman 2009-06-19-10-46

Liu,Hsinhan　劉鑫漢　2016-06-21-10-50

<a name="program0">
Prime, Prime Factor, Prime Gap.
This file is personal home work. Output
may contain error. Please verify first.

Box01:prime table; Box02:input integer; Box03:output factorized integer;
Box04:input integer factorization list, click [Prime counting] Box05:output summary

Box01   ; paste user prime list to Box01 or prime list

Build⇒ up to in Box01 Box13 comma
,List2Box09 ; Integer ends output to Box02 //a507061036
Box02   ;

example: , , , RUN ☞ ,
Box03   Box If integer factorization wrong, click [Build prime list]

Box04 box compare data in box03 and 04 ; box goto box07

Box04 in Box05 out
Box05   box goto box02

;
; List2Box09 QAboxc05.value='' ; Above factorize integer ; Below generate integer.
Box06 box, goto box02 why ASK⇒

Double prime gap ; ;
Output choice ;
Box07 box goto 02 ; [squeezed middle] alert here

Integer ends ; box goto 02 , ,
Box08 ; ; Box07 input Box08 output ,

no need input, output to Box09 Output may contain error !
Box09 ; ; output to

input already build primeArr[] output to Box09 //a507261629
primeArr[] array full range input ; Box09 output
Box08 zero line mean integer to prime summation correct. 22=19+3 to 22-19-3 get 0
Box08 none zero line mean integer - prime is not a prime. 22=13+0 to 22-13-0 get 9
QBboxc09.value='' ; box goto 02 ;
Above factorize integer to prime multiplication.
Below factorize integer to prime summation.
<a name=Goldbach>
above 22=2*11 ; below 22=11+11=17+5=19+3
German mathematician C. Goldbach (1690~1764) in his letter addressed
to Swiss mathematician L. Euler (1707~1783), Goldbach wrote:
Proposition (A) Every even integer ( ≥ 6 ) is the sum of two odd primes;
Proposition (B) Every odd integer ( ≥ 9 ) is the sum of three odd primes.
They were called Goldbach conjecture.
2016-06-19-09-47 Liu,Hsinhan access and 2016-06-25-15-55 copied
https://arxiv.org/ftp/math/papers/0309/0309103.pdf

Prime SUMMATION factorization random integer input to Box10 , output to Box11.
If next Odd subtract box number end with ';', Box11 output "no match primes+0"
Odd number - odd prime= white box number end with ';' or not?
Integer to prime summation. Click [prime sum random] input Box10, output Box11.
or Click [prime sum seq.] input Integer upper/lower bounds silver box, output Box11.
Box10 Prime sum random input ; , ,

U/L bounds Silver box number end with ';' or not?
Box11 Prime sum output ; Silver box input

Above silver box input output to Box10 and Box11
box goto 07 click [real number calculator] output to Box08.
Box12 debug Output may contain error !

QCboxc12.value=''; If silver box number end with ';', Box11 output even numbers only.

<a name=incrementalKount>
Integer to prime summation. Click [prime sum random] input Box13, output Box11.
or Click [prime sum seq.] input Integer upper/lower bounds silver box, output Box11.
Box13 is NOT an input box for [prime sum count]; Click [prime sum seq.], output to Box13.
Odd number - odd prime= white box number end with ';' or not?
Output long? or short? Output incremental long table, slower.
Dbounds box enter '10,346,5' start 10, end 346, increment 5 ; click [UStep]
Input next one line box , , , Output Box13.
Dbounds LightNavy box number end with ';' or not?
Box13 Integer output ; in Box13 ',' in Box01 newline

14=11+3 SUM, Box13 input NO output ; Box14 Box15 output
Box14 output ; , if silent

14=2*7 multiply Box13 input NO output ; Box14 Box15 output
Box15 output ; , if silent

Box16 output ; Box13 input Box16 output

function stringList_to_arrayList() dbg B16 in B17 out
Box17 output

,
Box18 output [mulToBox] dbg Y/N? Not output to Box18, faster.

Box19 output

Box20 output

QDboxc17.value=''; If LightNavy box number end with ';', Box14 output even numbers only.

<a name=primdraw01>
Drawing board zone one.
Graph area size, W: H:
x min: , x max: ; y min: , y max:
Above small drawing board. Below big drawing board. a505281617
Graph area size, W: H:
x min: , x max: ; y min: , y max: ;
Axes cross at x= , y= , Use x,y axis
1st line width= , color= , SoliDot=
2nd line width= , color= , SoliDot=
3rd line width= , color= , SoliDot=
4th line width= , color= , SoliDot=
5th line width= , color= , SoliDot=
6th line width= , color= , SoliDot=
7th line width= , color= , SoliDot= %A
8th line width= , color= , SoliDot=
Goldbach decomposition prime counting graph
Multiplication factors prime counting graph

Output may contain error ; ; ; ; ,
drawing board one, two, three, four Manual scale x,y this box is dummy.
 Prime Goldbach
; ; x:min/max ; y:min/max drawing board five
x axis is integer, y axis is prime count. Goldbach conjecture even=primeA+primeB. primeA=even-primeB
On x axis each primeA emit one curve, on that curve, even and primeB increase. If primeB step on a prime
primeA=7=20-13 local slope=1/1.
If primeB step on a none-prime,primeA=7=22-15, local slope=0/1.
 Prime Goldbach
prime summation decomposition RUN:☞ ; drawSum_a50719_upload();
10=7+3, 10=5+5 ;
Ebounds , ,
Choos$; ; ; Box21 input draw rainbow drawSum() prime multiplication decomposition RUN%☞ 120=2^3 * 3^1 * 5^1; intPrimMulCnt( ), drawMul( ) dbg Y/N? Not output to Box22, faster. pID=prime ID pID=0 primeArr[0]=2, pID=1 primeArr[1]=3 ... pID=6 primeArr[6]=17 Box07 Box22 output xygraph code ; QEboxd22.value='' ; ,  Box23 save Upper Left graph Box24 save Upper Right graph Save four boxes VML code for future reproduce. Click [save curve], read Box62. Paste VML code to Box91, then click [Draw Box91 VML code]  Box25 save Lower Left graph Box26 save Lower Right graph First VML code erased right after second draw. Box27 debug , , , , , Box28 debug ; Box29 debug Box30 debug QEboxd28.value='' ; ∂∂u/∂x/∂x+∂∂u/∂y/∂y=0 ; ∂∂v/∂x/∂x+∂∂v/∂y/∂y=0 ; ∂∂u/∂x/∂y , ∂∂v/∂x/∂y , ∂∂u/∂y/∂x , ∂∂v/∂y/∂x ; nArr=bounds_to_integerListArr(QEInp2Bound.value) nArr=stringList_to_arrayList(QEboxc13.value); <a name=primdraw02> Drawing board zone two. Graph area size, W: H: x min: , x max: ; y min: , y max: Above small drawing board. Below big drawing board. x_y Graph area size, W: H: x min: , x max: ; y min: , y max: ; %C 1st line width= , color= , SoliDot= 2nd line width= , color= , SoliDot= 3rd line width= , color= , SoliDot= 4th line width= , color= , SoliDot= 5th line width= , color= , SoliDot= 6th line width= , color= , SoliDot= 7th line width= , color= , SoliDot= %B 8th line width= , color= , SoliDot= Prime number drawing board Output may contain error ; 2016-07-12-18-47 Goldbach integer decomposition broken lines has three slopes. (1) slope=0/1 current even not contain primeA. Example primeA=11 at even 36-11=25 not prime. (2) slope=1/1 current even contain primeA one count. Example primeA=13 at even 36=23+13. (3) slope=2/1 current even contain primeA two count. Example primeA=17 at even 34=17+17. Interesting observation is that all prime jump at same count hight, show a horizontal wave. see graph gevn5000.jpg even 2 to 5000 and integer(even+odd) 2 to 2000 gint2000.jpg FRUN☞ Box31 input ; ; drawing board six drawSum_v59()  Prime Goldbach Click [BuildData to Box31], fill integer to Box31, click [sumSilent] click [draw Sum] Fbounds , Box31 input ; FRUN☞ drawSum_v59() ; Box32 output xygraph code ; QFboxd32.value='' ; function drawSum_v59() ,  Box33 save Upper Left graph Box34 save Upper Right graph  Box35 save Lower Left graph Box36 save Lower Right graph First VML code erased right after second draw. Box37 debug Box38 debug QFboxd38.value='' ; ∂∂u/∂x/∂x+∂∂u/∂y/∂y=0 ; ∂∂v/∂x/∂x+∂∂v/∂y/∂y=0 ; ∂∂u/∂x/∂y , ∂∂v/∂x/∂y , ∂∂u/∂y/∂x , ∂∂v/∂y/∂x ; <a name=primdraw03> Drawing board zone three. Graph area size, W: H: x min: , x max: ; y min: , y max: Above small drawing board. Below big drawing board. x_y Graph area size, W: H: x min: , x max: ; y min: , y max: ; %D Axes cross at x= , y= , Use x,y axis 1st line width= , color= , SoliDot= 2nd line width= , color= , SoliDot= 3rd line width= , color= , SoliDot= 4th line width= , color= , SoliDot= 5th line width= , color= , SoliDot= 6th line width= , color= , SoliDot= 7th line width= , color= , SoliDot= %E 8th line width= , color= , SoliDot= ; Even integer's two prime sum Output may contain error ; drawing board seven x axis is even integer ID, y axis is prime ID (y axis is NOT prime, y axis is NOT prime count). Goldbach conjecture even=primeA+primeB . For example 20=3+17=7+13, 20 meet 3,7,13,17; 20 not meet 5,11. Red bar is 6*n+1 type prime example 7=6*1+1, 31=6*5+1 etc. Blue bar is 6*n-1 type prime, example 5=6*1-1, 47=6*8-1 etc. One even number data in one column. If whole column only red bar, code draw thin red line over it. If column has only blue bar, draw blue line. even meet prime graph gap sequence=[] see horizontal line  Prime Goldbach Click [BuildData to Box41], fill integer to Box41, click [intPrimSumeet] click [drawSummet] Gbounds , GapSeq ; ; ; Output 1,2,3,4? 1 2 3 ; all GB41 ; , Box41 input ; more GRUN1☞ debug ? Box42 output intPrimSumeet 1 ; even Meet Prime ; 20: 3,7,13,17, ,List2Box47 ; , , function drawSummet( ) Box43 output intPrimSumeet 2 ; Prime Meet Even ; Box41,42,43,44 document see http://freeman2.com/tute0067.htm , local Box44 output intPrimSumeet 3 Prime 012 list 967: 111011011010011 Box45 ; QGboxd45.value='' Box46 ; QGboxd46.value='' Box47 debug input already build primeArr[] output to Box48 //a507261757 Box48 debug ; QGboxd48.value='' ; inArr=buildNumbArray(inArr) change input box string number to array number. function bounds_to_integerListArr(arg1) change [2,200] to 2,3,4,5,.....198,199,200 //a507161759 Display prime 6n±1 and prime gap Box50 ; QHboxd50.value='' ; <a name="a506232350"> Prime Counting Function Graph tute0058.htm#primeCountDraw click [A1] Prime Counting >> Square Counting ; ∑(1/Sq.)=π*π/6 , ∑(1/prime)→∞ see proof at http://freeman2.com/tute0058.htm#docA403 to docA454 Prime counting is dense than square counting. http://freeman2.com/primcnt1.png Prime is random? why prime march in pace? http://freeman2.com/gevn5000.jpg Odd number Goldbach decomposition has man-made factor. http://freeman2.com/gint2000.jpg User run this page has risk freeze your computer! http://freeman2.com/priminf0.jpg Prime is random? Why gint2000, gevn5000, gev10000.jpg have horizontal stripe? Draw even under 10000, after Build prime list to 10000. In other words, if build prime to 5000, NOT Draw even under 10000, simply not enough bricks. http://freeman2.com/gev10000.jpg <a name="docA001"> 2016-06-21-10-52 start 2016-06-16 Liu,Hsinhan begin write prime number program http://freeman2.com/prime_e1.htm upload 2016-06-23 prime_e1.htm is better than earlier http://freeman2.com/jsprime2.htm upload 2009-09-15 prime_e1.htm has following control buttons. <a name="docA002"> Build prime number list and display defined prime at box01.  Box01 ; paste user prime list to Box01 or prime list Build⇒ up to You can in Box01 Box02 ; Default:Program has 670 primes, from 2 to 5003 <a name="docA003"> 2016-06-21-11-11 here When open prime_e1.htm, program auto define prime from 2 to 5000 and display next message. Default:Program has 670 primes, from 2 to 5003 User can build prime to longer list. First fill a number in next click ASK⇒ message line show Build: Program has 348514 primes, from 2 to 5000011 To save time, program do NOT display prime table in Box01. User can ask program display current defined prime by clicking <a name="docA004"> Build prime list take few seconds time (Acer computer Aspire 5750Z-4830) User can copy from other source and paste existing prime list to Box01, click When click [Update], program read box01 value and save to primeArr[] future calculation will base on user's data. It is user's responsibility to supply correct prime table (start from 2) prime_e1.htm do not check this table is correct or not. User can do on purpose, supply wrong list and update to primeArr[] see what happen if prime table is wrong. To erase wrong table, click , program define 2 to 19. You need click to create longer list. 2016-06-21-11-31 here <a name="docA005">prime_link() Box02 is main input and Box03 is main output .  Box02 ; Default:Program has 670 primes, from 2 to 5003 example: , , , RUN ☞ , Box03 Box If integer factorization wrong, click [Build prime list] <a name="docA006"> 2016-06-21-11-48 here Box02 is input box, example input look like [[ 8 9 10 14 15 16 20 21 22 38 39 40 ..... ]] <a name="docA007"> If gap=4 click Box06 output above list. prime pair (13,17) has gap=4 and (13,17) squeezed (14,15,16), put squeezed list to Box02 and click RUN ☞ get [[ 8=2^3 9=3^2 10=2^1 * 5^1 14=2^1 * 7^1 15=3^1 * 5^1 16=2^4 20=2^2 * 5^1 21=3^1 * 7^1 22=2^1 * 11^1 38=2^1 * 19^1 39=3^1 * 13^1 40=2^3 * 5^1 ..... ]] <a name="docA008"> Same input, if click get [[ 8=2*2*2 9=3*3 10=2*5 14=2*7 15=3*5 16=2*2*2*2 20=2*2*5 21=3*7 22=2*11 38=2*19 39=3*13 40=2*2*2*5 ..... ]] This utility help you analyse if prime gap=4, the squeezed number pattern. prime_e1.htm allow prime gap=any even number. <a name="docA009"> Why ASK⇒ ? When program has Default:Program has 670 primes, from 2 to 5003 if click example fill 82003/50773 to Box02 and click get [[ 82003= 50773= ]] Because both are prime and greater than 5003. In this case, click add more prime to list solve problem. 2016-06-21-12-16 here <a name="docA010">prime_link() When program has Default:Program has 670 primes, from 2 to 5003 if click example fill 1512890160007 to Box02 and click get [[ 1512890160007=1229993^1 * 1229999^1 ]] Program auto build prime table to cover needs. <a name="docA011"> Auto build or not auto build, program exam sqrt(1512890160007), if sqrt(1512890160007) is greater than table maximum prime then auto build. sqrt(1512890160007)=1229995.9999963415 > 5003 example auto build. but sqrt(82003)=286.36 < 5003 example user build. Auto build or not auto build, if program exam 1512890160007 and build a table cover 1512890160007 it takes many hours. Liu,Hsinhan choose program exam sqrt(1512890160007) not exam 1512890160007. Because program has auto build prime table action, is needed. 2016-06-21-12-32 here <a name="docA012"> 2016-06-21-15-22 start Box04 and Box05 click buttons are next  Box04 box allow you compare two similar data in box03 and 04 Box04 in Box05 out Box05 box goto box02 ; Default:Program has 670 primes, from 2 to 5003 <a name="docA013"> 2016-06-21-15-33 here Box04 to Box05 main job is . Box04 input example is [[ 90=2^1 * 3^2 * 5^1 91=7^1 * 13^1 92=2^2 * 23^1 93=3^1 * 31^1 94=2^1 * 47^1 95=5^1 * 19^1 96=2^5 * 3^1 ]] <a name="docA014"> Box05 output example is next [[ 2 : 9 //prime 2 show up 9 times 3 : 4 //prime 3 show up 4 times 5 : 2 //same as above 7 : 1 11 : 0 13 : 1 17 : 0 19 : 1 23 : 1 29 : 0 31 : 1 37 : 0 41 : 0 43 : 0 47 : 1 ]] <a name="docA015">prime_link() Box04 input must use power expression. 90=2^1 * 3^2 * 5^1 is accepted Multiplication expression 90=2*3*3*5 do not work. Because code depend on '^' to find how many 3 and how many 5. <a name="docA016"> Box04 all numbers like "3^2" are considered. In other words, Box04 input represent what goal? squeezed numbers between (13,17) ? 14,15,16 numbers between (127,131) and (163,167) two four-gap prime set? 132,133,.....,161,162 Plain continuous integers 101,102, ... 199,200? These consideration are user's decision. summarize ALL "3^2" type numbers in Box04. <a name="docA017"> After write code, LiuHH paid attention that Box04 and Box05 set should shift to Box03 and Box04 set. Because Box03 output just 90=2^1 * 3^2 * 5^1 Later LiuHH did not shift, but add one line allow you compare two similar data in box03 and 04 If user has two similar power expression factorized integer numbers put side by side, it is easier to compare. 2016-06-21-15-59 here  <a name="docA018"> Box06 box, goto box02 Double prime gap ; ; Output choice ; Box07 box goto 02 ; [squeezed middle] alert here Integer ends ; box goto 02 <a name="docA019"> 2016-06-21-17-35 start Box06 and Box07 generate prime related integers. Input is Double prime gap This yellow box accept even numbers only. Start from 3 all prime are odd numbers. gap-2 prime pairs are (3,5), (827,829) etc. gap-4 prime pairs are (7,11), (379,383) etc. gap-6 prime pairs are (23,29), (557,563) etc. gap-8 prime pairs are (89,97), (683,691) etc. gap-100 prime pairs are (396733,396833) and (4783873,4783973) etc. <a name="docA020">prime_link() Input number 8 for example to yellow box. Use Program has 670 primes, from 2 to 5003 for test run. Because following buttons work cover primeArr[] array WHOLE RANGE. Too big array, wait longer time. If current is long Program has 348514 primes, from 2 to 5000011 Click change to then click , make sure see Build: Program has 670 primes, from 2 to 5003 You can work faster. <a name="docA021"> Click button , output to Box06 the next [[ 90 91 92 93 94 95 96 360 361 362 363 364 365 366 ..... ]] <a name="docA022"> Gap-8 primes are (89,97), (359,367) etc. (89,97) squeeze 90,91,92,93,94,95,96 . (359,367) squeeze 360,361,...,365,366 . Box06 list those squeezed numbers. Click send data to Box02. Next click study factorization pattern. <a name="docA023"> Below Box06 click button Box06 and Box07 both output answer. Box06 has next [[ 89,97 359,367 389,397 401,409 449,457 479,487 491,499 683,691 ..... ]] <a name="docA024"> Box07 has next [[ 97,359 367,389 397,401 409,449 457,479 487,491 499,683 691,701 ..... ]] <a name="docA025">prime_link() Box06 output (89,97), (359,367) they are gap-8 pair. Box07 output (97,359), (367,389) they are gap-8 pair outsider bounds. If someone want study gap-8 pair outsider numbers. Bounds are in Box07. If you want study gap-8 pair outsider numbers between 499,683 . Copy and paste 499,683 to Integer ends Replace 101,200 with 499,683 then click and click . Continue Box02 procedure. <a name="docA026"> Below Box06 click button Box07 output [[ Double MIDDLE, gap=8. Number +/-4 are prime numbers. 93 363 393 405 453 483 495 ..... OUTSIDE-gap MIDDLE, gap=8 228 378 399 429 468 ..... ]] gap-8 first pair is (89,97), its middle is 93. gap-8 2nd pair is (359,367), its middle is 363. gap-8 3rd pair is (389,397), its middle is 393. (89,97) to (359,367) middle point is (359-97)/2+97=228 which is OUTSIDE-gap MIDDLE generate all middle numbers, help user analysis. <a name="docA027"> Below Box06 click button follow , that is click first, second. work with green box Output choice Green box read yellow box value. If Double prime gap has value 8, then green box choice from 1 to 7. Click green box, box value change below 8. <a name="docA028"> then gap-8 prime pairs 89,97 ; 359,367 ; 389,397 ; 401,409 ; 449,457 squeezed numbers are 90, 91, 92, 93, 94, 95, 96 360,361,362,363,364,365,366 390,391,392,393,394,395,396 402,403,404,405,406,407,408 450,451,452,453,454,455,456 etc If green box has number 2 , click [squeezed number] then [squeezed middle] all second number recorded in Box06. They are 91,361,391,403,451,481,493 ... <a name="docA029"> move Box06 output to Box07 output to Box06 the next [[ 90 91 92 93 94 95 96 91 361 391 403 451 481 493 685 703 721 745 763 913 931 985 1111 1165 1195 1375 1441 1525 1561 1573 1735 1825 1981 2005 2155 2215 2245 2275 ..... ]] <a name="docA030">prime_link() Top seven number is first squeezed set 90,91,92,93,94,95,96 The following numbers are NOT a whole set. The following numbers are second number in each set. First few are 91 361 391 403 451 ..... This function help reader study each gap-8 second number has what pattern. Initially, LiuHH paid attention to middle only. No choice, later think it is easy to add choice code. <a name="docA031"> Below Box06 click button If yellow box has value 8. output to Box06 the next [[ 88 <== outsider 89 <== prime 90 91 92 93 <== squeezed middle 94 95 96 97 <== prime 98 <== outsider 358 <== outsider 359 <== prime 360 361 <== squeezed 362 363 364 365 366 367 <== prime 368 <== outsider ..... ]] <a name="docA032"> [squeezed+prime] is actually squeezed+prime+twoNeighbor LiuHH just hope see prime factorized numbers what pattern they have. For example gap-8 middle they all have factor 3. gap-10 middle they all have factor 6. gap-12 middle they all NOT have factor 6. These observation do not help much. a506231921 Liu,Hsinhan did not see any exciting pattern. prime_e1.htm is online If many scholar study prime pattern, there are greater chance find exciting pattern. <a name="docA033"> Now http://freeman2.com/prime_e1.htm is in your hand. If you want to analysis prime factorized numbers, prime_e1.htm is a tool you can use. As usual No one proofread Liu,Hsinhan work No one test run Liu,Hsinhan program. This file may contain error. Reader must use prime_e1.htm with caution. Liu,Hsinhan 劉鑫漢 2016-06-21-19-42 stop <a name="docA034"> 2016-06-21-22-22 start 2016-06-21-22-10 LiuHH run (Build spend 30 seconds) Build prime list up to 20000000 Build: Program has 1270608 primes, from 2 to 20000003 Double prime gap 100 Double MIDDLE, gap=100. Number +/-50 are prime numbers. [[ 396783=3^2 * 44087^1 838299=3^1 * 7^1 * 11^1 * 19^1 * 191^1 1313517=3^1 * 41^1 * 59^1 * 181^1 1648131=3^1 * 83^1 * 6619^1 1655757=3^2 * 183973^1 ..... 19294473=3^1 * 11^1 * 17^1 * 163^1 * 211^1 19304217=3^3 * 714971^1 19416561=3^1 * 6472187^1 19871331=3^1 * 37^1 * 179021^1 19917117=3^3 * 11^1 * 67061^1 ]] all middle have factor 3, no factor 2. 2016-06-21-22-28 stop <a name="docA035">prime_link() 2016-06-29-21-05 start Update 2016-07-01 added Goldbach conjecture 2016-06-19-09-53 Liu,Hsinhan access http://www.ijpam.eu/contents/2013-84-1/3/3.pdf [[ As is known, on June 7, 1742, Christian Goldbach in a letter to Leonhard Euler, Goldbach argued that, every even natural number greater than 4 can be written as a sum of two primes, namely: 2n = p + q, where n > 2, and p, q are prime numbers. ]] Liu,Hsinhan added a Goldbach section Box10, Box11 and Box12. Goldbach section has next structure.  <a name="docA036"> Odd number - odd prime= white box number end with ';' or not? Box10 Prime sum random input ; , , Integer upper/lower bounds Silver box number end with ';' or not? Box11 Prime sum output ; Silver box input Above silver box input output to Box10 and Box11 box goto 07 click [real number calculator] output to Box08. Box12 debug Output may contain error ! <a name="docA037"> 2016-06-29-21-35 here Why need Odd number - odd prime= white box number end with ';' or not? Goldbach conjecture say that Every even integer ( ≥ 6 ) is the sum of two odd primes; Every odd integer ( ≥ 9 ) is the sum of three odd primes. In fact odd integer ride on even integer. Because subtract a prime 3, odd integer 25 become even 22, subtract a prime 5, odd integer 25 become even 20, subtract a prime 7, odd integer 25 become even 18, subtract prime 11 , odd integer 25 become even 14. Larger odd integer, more alternative choice it has. <a name="docA038"> Program can force all odd integer subtract prime 3 But this Odd number - odd prime= box allow user decide what prime number be a factor of odd integer. If user assign 19 to white box. All odd integer in Box10 will have 19 as a factor. If input Box10 has odd integer 15, which is less than 19. Program use 3 as a participate prime. 2016-06-29-22-07 stop <a name="docA039"> 2016-06-30-10-05 start If user input 29 to and in Box10 Prime sum random input 3 15 21 33 35 assign numbers they are both greater than and less than 29. Number 35 subtract user assigned prime 29 get 35-29=6 The even integer 6 is just enough for Goldbach split to sum of two primes that is 6=3+3. <a name="docA040">prime_link() Odd integer 35=3+3+29 satisfy Goldbach conjecture. Other odd integers in Box10 have trouble. Odd integer 33 subtract 29 get 4 Four is less than Goldbach requirement six. Odd integer 21 subtract 29 get -8 Negative number is not a concern in integer prime summation decomposition analysis. For this consideration. If user input prime P0 in white box and input odd integer I0 to Box10. If I0>=P0+6 I0 decomposition include P0. If I0< P0+6 I0 decomposition include prime 3. 2016-06-30-10-32 here <a name="docA041"> White box and are different. 2016-06-30-10-40 cannon shock. 2016-06-30-10-41 cannon shock. If prime 3 in white box follow a ';', then Box11 output [[ 34=31+3 34=29+5 34=23+11 34=19+0 <== no match line 34=17+17 35=31+0 <== no match line 35=29+3+3 35=23+0 <== no match line 35=19+13+3 ..... ]] <a name="docA042"> ';' in white box, bring "<== no match line " to Box11. This is valid for both and 2016-06-30-10-49 here <a name="docA043"> Box10 is input box for Integer upper/lower bounds is input box for read Box10 value and change integer to sum of two prime or sum of three primes. Box10 value allow random order integers. read silver color box value. Need two integers. One begin number, one end number. Code in click button send sequential integers to Box10 and send prime summation decomposition answer to Box11. Integer prime multiplication decomposition is unique. Integer prime summation decomposition has multiple answers. <a name="docA044"> Integer upper/lower bounds number end with ';' or not , program behave differently. If no ';' in Click Box10 and Box11 output odd integer and even integer. If in has ';' Click Box10 and Box11 output even integer only. Silver box has ';' or not only read silver box. NOT read silver box. 2016-06-30-11-00 here <a name="docA045">prime_link() 2016-06-30-14-32 start Below Box11, there is a click button This button input from silver box, output to Box11 and Box12. Assume silver box has [6,20], send number sequence to Box10. [[ 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ]] <a name="docA046"> function GoldbachCount(arg1) receive above input numbers. For even number GoldbachCount(arg1) do prime multiplication decomposition and do prime summation decomposition. For example 22=2*11 //prime multiplication decomposition 22=19+3 ; 22=17+5 ; 22=11+11 //summation decomp. <a name="docA047"> output to Box11 [[ 6, 1,1,1 8, 1,2,1 10,2,3,0 12,1,2,1 14,2,3,1 16,2,4,0 18,2,4,1 20,2,4,2 ]] All odd number in Box10 ignored in Box11. Because odd number prime summation decomposition is just replay for even numbers. No need repeat. <a name="docA048"> Box11 "14,2,3,1" line say Input number 14 prime summation decomposition 14=13+0 <== inactive equation, one prime 13 14=11+3 <== active equation, two primes 11,3 14=7+7 <== active equation, one prime 7 14 has 2 active equations 3 active primes 11,7,3 14 has 1 inactive equations 1 inactive prime 13 above 14=13+0 is inactive equation, because 13 cannot sum any other prime to get 14. Inactive equation count is same as inactive prime count. <a name="docA049"> Box11 has four columns, left most is input number. Box12 has three columns, left most is prime number. Box12 output next [[ 2:0,15 3:7,4 5:6,2 7:6,1 11:3,0 13:3,0 17:1,0 ]] <a name="docA050">prime_link() Box12 line "7:6,1" say the following Input Box10 all EVEN numbers prime summation decomposition prime 7 has 6 count. prime multiplication decomposition 7 has 1 count. Below Box04, click button do not refuse odd numbers. Box05 output prime multiplication decomposition count MAY include odd integers. Below Box11, click button refuse odd numbers. Box12 output not include odd integers. 2016-06-30-15-29 stop <a name="docA051"> 2016-06-30-17-43 start After 2016-06-23 upload http://freeman2.com/prime_e1.htm added the following boxes.  Box07 Integer ends ; box goto 02 Box08 ; ; Box07 input Box08 output , no need input, output to Box09 Output may contain error ! Box09 ; ; output to Box09 QBboxc09.value=''; ; ; box goto 02 ; <a name="docA052"> 2016-06-30-17-48 here button add "2,2000" to Integer ends [example⇒] do not add long data to bigger [textarea] box. If click program output integer list to Box08 start from 2 end at 2000 build increase sequence. On the other hand Integer upper/lower bounds build increase and build decrease sequence. <a name="docA053"> 2016-06-24-10-09 (a506241009) Liu,Hsinhan added to prime_e1.htm to solve real number calculation problem. Input at Box07 output to Box08. This real number calculator has function compatible to prime_e1.htm Click program write next to Box07 [[ 160392960=2*2*2*2*2*2*2*2*3*3*3*3*5*7*13*17 103=47+53+3 102=89+13 2=2 ]] <a name="docA054"> Click Box08 output answer [[ 160392960-2*2*2*2*2*2*2*2*3*3*3*3*5*7*13*17 0 103-47-53-3 0 102-89-13 0 2-2 0 ]] <a name="docA055">prime_link() Program change '=' to '-' and change '+' to '-' allow user verify first input line 160392960=2*2*2*2*2*2*2*2*3*3*3*3*5*7*13*17 both side are equal and confirm third input line 89+13 indeed sum to 102, because 102-89-13=0 <a name="docA056"> If input 102=89+13*1 '+' and '*' show up at same line, auto conversion stop work. Click program write code lines to Box07. Click Box08 output regular real calculator answer. <a name="docA057"> Click program output to Box09 [[ maxDif=34; maxID=217; Max gap occur at prime=1327,1361 1 <== difference of 2,3 2 <== difference of 3,5 2 <== difference of 5,7 4 <== difference of 7,11 2 <== difference of 11,13 4 <== difference of 13,17 etc. 2 4 6 2 6 4 2 ..... ]] <a name="docA058"> Before you run full range prime difference, you may need to know how many stored in primeArr[]. Click read Box09 first line. If fill two numbers in and click code output to Box09 prime difference in this limited range. Prime difference list help you understand the prime density and distribution condition. <a name="docA059"> output full range prime and middle integer between two primes. output full range middle integer between two primes. Liu,Hsinhan 劉鑫漢 2016-06-30-18-39 stop <a name="docA060">prime_link() 2016-06-30-21-17 start 2016-06-30-21-00 created button. Click [ex0] program fill javascript code to Box07. Click build square sequence in Box08. Box08 first line is 'oupStr;' delete 'oupStr;' copy only integers to Box01, Click button let primeArr[] store squares. <a name="docA061"> Use prime as building block, factorize integer to prime multiplication and factorize integer to prime summation all success. Use square as building block, factorize integer to square multiplication and factorize integer to square summation all fail. Because prime is dense and square is sparse. See Prime Counting Function Graph http://freeman2.com/tute0058.htm#primeCountDraw click [A1] http://freeman2.com/primcnt1.png 2016-06-30-21-27 stop <a name=a507070052> 2016-07-07-00-52 Acer1 computer mouse pointer disappear two minutes, computer don_hi-don_lo noise 7 or 8 times. a507070052 <a name="docA062"> 2016-07-13-09-21 start update 2016-07-13 prime_e1.htm added drawing capability. Drawing board 6 produced the graph uploaded to http://freeman2.com/gevn5000.jpg Goldbach even 5000 http://freeman2.com/gint2000.jpg Goldbach int. 2000 After upload gevn5000.jpg , Liu,Hsinhan continue improve drawing code and get Drawing board 1,2,3,4,5 2016-07-11 LiuHH find 'improved' drawing board can not draw points greater than 500. But earlier file 07/08/2016 09:41 PM 551,166 prime_e1_v59_first_draw.htm can draw 2500 points ! What is the trouble? LiuHH included 2016-07-08 drawing code to prime_e1.htm 07/11/2016 08:12 PM 661,734 prime_e1_v68_drawSum_v59().htm and name this older code as Drawing board 6 <a name="docA063"> Drawing board 6 has next control boxes/buttons  Graph Message ; Fill integer to Box31, click [sumSilent] click [draw Sum] Fbounds , Box31 input ; RUN ☞ ; <a name="docA064"> 2016-07-13-09-51 here Drawing board 6 read input data from Box31. User can copy paste integers to Box31. Integers should be positive and increase order. Green box and button help you build input data. Left green box define integer begin, integer end and step size. In most case use 2,300 for faster draw. In "2,5001", 2 is first in-range integer. 5001 is first out-range integer. LiuHH's Acer1 computer draw "2,5001" continuous integer graph. But vary gap integer under 5000 fail. <a name="docA065">prime_link() Right green box has the following six choices 1:=integer [Fbounds] continuous integer 2 to 5000 2:=prime in [Fbounds] prime under 5000 3:=pr Middle [Fbounds] prime middle under 5000 4:=Pr+Middle [Fbounds] prime+middle under 5000 5:=even in [Fbounds] even number under 5000 6:=step in [Fbounds] step number under 5000 Prime 43 and prime 47 has middle (43+47)/2=45 prime middle list 2.5,4,6,9,12,15,18,21,26, ... prime+middle list 2,2.5,3,3,4,5,5,6,7,7,9,11... <a name="docA066"> If left green box has three numbers 2,5000,8 If right green box has 6:=step in [Fbounds] Click [BuildData to Box31] Box31 list 2,10,18,26,34,42,50,58,66,74,82,..... Green box and [BuildData to Box31] are utilities help user build simple data. Assume build Box31 data as next prime middle 4,6,9,12,15,18,21,26,30,34,39,42,45,50,56,60,64,69,72,76,81,86,93, <a name="docA067"> next click program produce [[ 4=2+2 6=3+3 9=3+3+3 <= integer 9, prime 3 has three count 12=7+5 15=7+5+3 18=13+5 18=11+7 21=13+5+3 21=11+7+3 26=23+3 26=19+7 26=13+13 30=23+7 30=19+11 30=17+13 34=31+3 34=29+5 34=23+11 34=17+17 39=31+5+3 39=29+7+3 39=23+13+3 39=19+17+3 ..... ]] <a name="docA068"> prime incremental summation is next 2,3,5,7,11 ..... [[ | | | | | 4: 2 | | | | 6: 2,2 | | | 9: 2,5 | | | <= to integer 9, prime sum to 5 count 12: 2,5,1,1 | 15: 2,6,2,2 | 18: 2,6,3,3,1,1 21: 2,8,4,4,2,2 26: 2,9,4,5,2,4,0,1,1 30: 2,9,4,6,3,5,1,2,2 34: 2,10,5,6,4,5,3,2,3,1,1 39: 2,14,6,7,4,6,4,3,4,2,2 42: 2,14,7,7,5,7,4,4,5,3,3,1 45: 2,18,8,7,6,8,4,5,6,4,4,2 ..... ]] <a name="docA069"> Click drawing code draw prime 3 curve (6,2) (9,5) (12,5) (15,6) ..... prime 5 curve (12,1) (15,2) (18,3) (21,4) ..... ..... Many think that prime show up randomly, but http://freeman2.com/gevn5000.jpg suggest that all prime same_jump at same_number_of_counts. gevn5000.jpg horizontal wave/wrinkle is a clue. 2016-07-13-10-52 here <a name="docA070">prime_link() Below Drawing board 6 , there is a message line Graph Message ; Fill integer to Box31, click [sumSilent] click [draw Sum] If click [sumSilent] message change to x-axis max. =498 , y-axis max. =25, second maxY=24. Adjust x/y min/max below 'x_y'. If click [draw Sum] message change to [draw Sum] Begin=2016-07-13-09-19-45.192, End=2016-07-13-09-19-56.717: 11.525 sec <a name="docA071"> Drawing board 6 need manual change x/y min/max If use then x maximum should be 5000 [sumSilent] message suggest x maximum and y maximum x-axis max. =1999 , y-axis max. =27060, second maxY=602. Adjust x/y min/max below 'x_y' If input has even only, y-axis max. and second maxY are the same. If input has odd number, y-axis max. is much greater than second maxY. Because all odd number has prime 3 39=31+5+3 39=29+7+3 39=23+13+3 39=19+17+3 <a name="docA072"> In odd=even+3 the y-axis max. =27060, second maxY=602. First maximum and second maximum have big difference. In this case follow second maxY=602, in y max:[ box ] fill 600. Ignore prime 3 first maximum=27060 . is a button work and created gevn5000.jpg [draw Sum] still fail draw prime middle < 5000 is a button attempt find out why program freeze forever. Not done yet. 2016-07-13-11-28 stop <a name="docA073"> 2016-07-13-15-30 start Above is Box31 and drawing board 6. Below is Box21 and drawing board 1,2,3,4,5. drawing board 1,2,3,4,5 CHANGED RUN:☞ ; ; ; function drawSum( ); Ebounds , , ⇐ active Choose Box21 input Graph may be ERROR! , Box22 output xygraph code ; QE0boxd22.value='' ; , <a name="docA074"> 2016-07-13-15-38 here and initially are two separate buttons. Later [Draw yellow box] included [sumSilent] function. Now click [Draw yellow box] code read all yellow boxes and run [sumSilent] and draw at once. Ebounds specify input data begin, end and step values. Code build input data to Box31. If you have data in hand and not need code build data, in second box choose "7:=Box21 data as is" <a name="docA075">prime_link() Next data type box has next choices. 1:=integer [Ebounds] 2:=prime in [Ebounds] 3:=pr Middle [Ebounds] 4:=Pr+Middle [Ebounds] 5:=even in [Ebounds] 6:=step in [Ebounds] 7:=Box21 data as is Click box and choose one value. is one click for all button. If you do not want [Draw yellow box] build data, choose "7:=Box21 data as is" Compare with Box31 input RUN=☞ user must click to build data, green box do not have "7:=Box21 data as is" <a name="docA076"> Next yellow box attempt auto change drawing board 5 x/y min/max range. Free user from worry x/y scale. This is LiuHH's attempt, not work perfect now. Other choice is "Manual scale x,y" <a name="docA077"> Next yellow box choose one drawing from five possible boards. Four smaller board and one bigger board. Five board allow user put graph side by side compare. <a name="docA078"> Next yellow box choose output method. Choices are Graph, no VML VML, no Graph SLOW,Graph & VML <a name="docA079"> Next silver box When code is done, silver box will change to yellow box. Choices are Goldbach Prime sum 14=11+3 Prime multiplication 14=2*7 Log Prime multiplication <a name="docA080">prime_link() Today 2016-07-13-16-18 main work is integer's prime summation decomposition. integer's prime multiplication decomposition is future work. sum. decomp. all prime show up nearly equal. mul. decomp. prime 2,3 show up higher than other. For this consideration, "Log Prime multiplication" is a choice. "⇐ active" point to yellow boxes, compare with silver box, which is inactive. <a name="docA081"> change board five x/y min/max range such that it is possible zoom in gevn5000.jpg graph obstacles in greater detail. Not work now 2016-07-13-16-28. is debug button try find out why prime_middle has infinite loop and find out how to write zoom in code. <a name="docA082"> 2016-07-11-17-15-40 re-produce gevn5000.jpg 2016-07-11-17-16-20 get graph in 40 seconds Liu,Hsinhan's computer is Acer computer Aspire 5750Z-4830. MSIE 9.0 Version:9.0.8112.16421 gevn5000.jpg draw even number under 5000. Total 5000/2=2500 points. Each time draw gevn5000 OK. Twice draw even number under 10000. Both time MSIE VML stop work after one minute. not freeze screen. <a name=a507151415> 2016-07-15-14-15 realize why "MSIE VML stop work after one minute" Because prime_e1.htm define prime up to 5000. When run 10000, LiuHH forget to define more prime numbers. 2016-07-15-14-21 re-do "draw even number under 10000" save graph as http://freeman2.com/gev10000.jpg 07/15/2016 03:35 PM 439,850 gev10000.jpg 07/09/2016 05:42 AM 94,382 gevn5000.jpg 07/09/2016 11:10 AM 115,204 gint2000.jpg gevn5000.jpg and gint2000.jpg are compressed. gev10000.jpg keep bigger file size for better resolution. Because gev10000.jpg draw to 10000. 2016-07-15-15-50 record <a name="docA083"> Many times draw prime middle for prime less than 5000 There are 669 middle points. For example prime 19 and 23 has middle point (19+23)/2=21 Each time run prime middle always run one hour and freeze MSIE. See screen shot http://freeman2.com/priminf0.jpg User run this page has risk freeze your computer! 2016-07-13-16-46 stop <a name="docA084"> 2016-07-15-16-16 start Update 2016-07-15 explain why "MSIE VML stop work after one minute" 2016-07-15-14-21 re-do "draw even number under 10000" upload to http://freeman2.com/gev10000.jpg 2016-07-14-20-07 use YTD download https://www.youtube.com/watch?v=PtsrAw1LR3E https://www.youtube.com/watch?v=yPrbArbhPxk https://www.youtube.com/watch?v=pO7Egc5Dtqs <a name="docA085">prime_link() 2016-07-15-09-11 play Math Mornings at Yale_ The Patterns in the Primes, with Andrew Granville.mp4 1:00:14 2016-07-15-10-47 done 2016-07-15-10-49 play Terence Tao_ Structure and Randomness in the Prime Numbers, UCLA.mp4 47:50 2016-07-15-13-42 done 2016-07-15-13-44 play Introduction to Higher Mathematics - Lecture 10_ Number Theory.mp4 2016-07-15-18-39 done <a name="docA086"> 2016-07-15-14-15 realize why "MSIE VML stop work after one minute" 2016-07-15-14-21 build gev10000.jpg 2016-07-15-16-24 record 2016-07-15-16-52 upload gev10000.jpg <a name="docA087"> 2016-07-19-20-45 update 2016-07-19 added Box41 input and output to Box42 Box43 Box44. Input is even integers 6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,... Output is Box42 output even Meet Prime, example 20: 3,7,13,17, Box43 output Prime Meet Even, example 991: 994,996,998 Box44 output Prime 012 list , example 967: 111011011010011 Liu,Hsinhan main goal is Prime 012 list. After create gint2000.jpg , gevn5000.jpg , and gev10000.jpg graph show horizontal stripe which is a puzzle. With in hand, horizontal stripe mystery solved, see http://freeman2.com/tute0067.htm#docA019 http://freeman2.com/tute0067.htm#docA043 http://freeman2.com/tute0067.htm#tute0067_index 2016-07-19-20-59 stop <a name="docA088"> 2016-07-28-15-33 start Update 2016-07-28 improved code efficiency. Earlier version if input non-prime 33 to Odd number - odd prime= If primeArr[] stored million primes, program find 33 all the way to list end. But when search passed 37, future data always greater than 33. No need search future data. Similiar trouble exist in other place. Improved code stop search after prime is greater than integer. <a name="docA089"> Update 2016-07-28 added above Box02, Box06, Box42. Define primeA6n[] but no output to save time. Added below Box09 output to Box09 and below Box47 output to Box47. <a name="docA090">prime_link() Update 2016-07-28 added drawing board seven graph function. Click program fill data to Box41. Next click let program define summet array. To see the structure of summet goto prime_e1.htm Box48, click After click define summet, next click , wait for 10 seconds to 40 seconds to get graph like http://freeman2.com/gevmpri0.jpg //spend 10 seconds http://freeman2.com/gevmpri1.jpg //spend 35 seconds <a name="docA091"> Drawing board seven graph mark 6*n+1 type prime with red vertical bar, mark 6*n-1 type prime with blue horizontal bar. If one even number meet ONLY red bar program draw a thin red vertical solid line over them. If one even number meet nearly all red bar except top most be a blue bar, program draw a thin red dot line over them. If one even number meet ONLY blue bar, program draw a thin blue solid line over them. Prime 3 cannot fit 6*n+1 or 6*n-1 , program define 3 be red bar. Prime 5 is first prime fit 6*n-1 , program draw prime 5 be blue bar. Thin red vertical solid line and thin blue solid line worth our attention. See more about gevmpri0.jpg and gevmpri1.jpg at http://freeman2.com/tute0067.htm#docA052 2016-07-28-16-10 stop <a name="docA092"> 2016-07-28-21-45 start Update a507282055 = Update 2016-07-28-20-55 2016-07-28-20-55 made code correction, allow program draw blue dot lines. Blue dot line mean on that line most prime are 6*n-1 type only very top and very bottom are 6*n+1 type. Earlier version program can draw red dot line, but was unable to draw blue dot. Now identify blue dot lines help reader think prime structure. 2016-07-28-21-50 stop <a name="docA093"> 2016-08-05-16-33 start Update 2016-08-06 added prime multiplication decomposition RUN%☞ Drawing board 1,2,3,4,5 has three color groups. Yellow Goldbach conjecture Red Goldbach conjecture Green prime multiplication decomposition Control arrangement is next.  prime summation decomposition RUN:☞ ; drawSum_a50719_upload(); 10=7+3, 10=5+5 ; Ebounds , , Choos$ ; ; Goldbach conjecture
Box21 input draw rainbow drawSum()

prime multiplication decomposition RUN%☞ ,
120=2^3 * 3^1 * 5^1; intPrimMulCnt( ), drawMul( ) dbg Y/N? Not output to Box22, faster.
<a name="docA094">
2016-08-05-16-50 here
Button  call drawSum()
Button  call drawMul()
prime summation decomposition example 10=7+3, 10=5+5
prime multiplication decomposition 120=2^3 * 3^1 * 5^1

Initially, drawSum() has one version.
2016-07-08-23-25 drawSum_v59() created gevn5000.jpg
Later version drawSum() cannot draw graph, trap in
has graph capability. Liu,Hsinhan continue correct
drawSum() but was unable to find why draw 2 to 5000
get graph, draw 2 to 300 get graph, draw 2 to 1000
can never get graph. This problem is not solved.
Final version drawSum() draw 2 to 1000 limit to
primeID=1 to 10 get graph for integer up to 1000.
primeArr[167]=997 There are 168 primes under 1000.
Program draw 10/168 reduce load and show 10 curves.
Look like rainbow, and mark "draw rainbow" If user
click  number auto increase.
To decrease number, highlight and type number.

<a name="docA096">
Yellow  read five yellow boxes, and do as
box show.
control input integer start end value.
Step value is used only if ask
select data type. Default [1:=integer]
To draw  use data type [1:=integer]
To draw  use data type [5:=even ]
If draw  use data type [1:=integer ] each
odd contribute several prime3, and prime3 curve shoot
to sky immediately.
If draw red  [DrawSum %%] read all yellow

<a name="docA097">
If draw green  [DrawMul()] read all yellow
green box
If draw prime multiplication decomposition from 2 to
1000. Each even has one prime2. 64 has prime2 six
times then prime2 curve value is higher than other
prime curve. For example integer 2 to 1000
//[mulToBox] Integer prime mul, prime total count.
2: 994
3: 498
5: 249
7: 164
11: 98
13: 81
17: 61
.....
971: 1
977: 1
983: 1
991: 1
997: 1

<a name="docA098">
In this case if change green box from
'Prime multiplication 14=2*7'
to 'Log Prime multiplication'
Output curve is log scale and may look clearer.
Sample output
http://freeman2.com/pmul300a.jpg  //Y axis is linear
http://freeman2.com/pmul300b.jpg  //Y axis is log()
pmul300a.jpg 2016-08-03-21-58-57.170 elapse=54.327 sec.
pmul300b.jpg 2016-08-04-01-31-48.277 elapse=53.21 sec.
2016-08-05-17-55 stop

<a name="docA099">
2016-08-05-19-36 start
Below Box07 find ♫ , click  program fill code
to Box07, click  get output at
Box08. Prime ID range from 0,1,2,3,4,5 ... to what
ever defined. primeArr[pID] output prime number.
pID can be 0,2,4,6,8 even, for pID>0 primeArr[pID]
always be odd prime number.
Another file http://freeman2.com/prime_e2.htm#bx07
same  button, also supply
primeA6n[pID] //0 if prime=6*n-1; 1 if prime=6*n+1
prime_e1.htm start up with prime defined to    5003
prime_e2.htm start up with prime defined to 1000003
prime_e2.htm also define primeA6n[] and primeGap[]
2016-08-05-19-47

<a name="docA100">
2017-03-28-16-16
Update 2017-03-29 add horizontal line in
even meet prime graph
Array primeArr[] store prime, use primeID (pID)
as index. First few are
pID   :  0,  1,  2,  3,  4,  5,  6,  7,  8,...
prime :  2,  3,  5,  7, 11, 13, 17, 19, 23,...
primeArr[2]=5, primeArr[8]=23 etc.
In even meet prime graph y axis is primeID (pID).
Choose pID that is because save space in vertical
direction. On the other hand, if y axis draw prime
within same vertical space, graph contain less data.
y axis use pID it has disadvantage. If use y=prime
then consecutive 197, 199, 211 show gap in scale. If
use y=pID, prime 197, 199, 211 become pID 44,45,46
gap scale lost.
<a name="docA101">
2017-03-24-21-09 add 
GapSeq ; ; ;
2017-03-28-16-45 here
If fill [4,2] to above box, ask program mark
gap sequence 4,2 Output will draw a purple
horizontal line. Consecutive prime set
7, 11, 13 and 13, 17, 19 and 37, 41, 43 all
have gap sequence 4,2. First primeID 3,5,11
has a purple horizontal line. pID 3=prime 7,
pID 5=prime 13, pID 11=prime 37. This function
compensate "gap scale lost" problem.
2017-03-28-16-57

<a name="docA102">
2017-03-28-21-36
To draw http://freeman2.com/gevmpri1.jpg
in even meet prime graph click buttons
click the following
set up graph x,y range
create even numbers
build graph data
in prime_e1.htm draw gevmpri1.jpg
User should get gevmpri1.jpg LiuHH drawing time is next
[drawSummet] Begin=2017-03-28-21-47-03.064, End=2017-03-28-21-47-36.612: 33.548 sec
When read gevmpri1.jpg , you may need next table
http://freeman2.com/prim6n01.htm
In prim6n01.htm 6*n-1 prime are blue bars in gevmpri1.jpg
In prim6n01.htm 6*n+1 prime are red  bars in gevmpri1.jpg
2017-03-28-21-56


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<a name=JavascriptIndex>
Javascript index
http://freeman2.com/jsindex2.htm   local
Save graph code to same folder as htm files.
http://freeman2.com/jsgraph2.js   local

Older Prime number and prime decomposition
http://freeman2.com/jsprime2.htm