First P.I.CO file   Second   jsprime2.htm
Pair Prime number drawing board
User run this page has risk freeze your computer!
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   ★★★★★
Please use MSIE browser to read this file.
Program environment Acer computer Aspire 5750Z-4830.
MSIE 9.0 Version:9.0.8112.16421
This file is personal home work. No one
proofread. Cannot promise correctness.
If you suspect any view point wrong,
please ask a math expert near by.
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.


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a
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Graph alert display here.
; ; 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.
P
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G
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a
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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.

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()
P
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G
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a
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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
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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">
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">
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">
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">
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">
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">
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">
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 

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">
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">
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">
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">
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">
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">
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">
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">
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">
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">
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">
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_a50719_upload()
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

<a name="docA095">
Initially, drawSum() has one version. 
2016-07-08-23-25 drawSum_v59() created gevn5000.jpg 
Later version drawSum() cannot draw graph, trap in 
infinite loop. 2016-07-19 drawSum_a50719_upload() 
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 
box and read red box  

<a name="docA097">
If draw green  [DrawMul()] read all yellow 
box and read red box  and read 
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
primeGap[pID] //gap follow primeA; primeA+gap=primeB
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

[=][=]

<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
jsprime2.htm first upload 2009-09-15


Prime number and prime decomposition
http://freeman2.com/prime_e1.htm
Chinese version 素數函數及整數之素數分解
http://freeman2.com/prime_c1.htm
prime_e1.htm first upload 2016-06-23

Thank you for visiting Freeman's page. 
Freeman Liu,Hsinhan 劉鑫漢 2016-06-21-10-35

Please visit freeman2.com/jspico_e.htm
Please get prime list: http://freeman2.com/prime1m2.htm