Prime number study notes tute0068
Liu,Hsinhan present a thinking problem to general public. 2017-04-16-18-03
Even meet Prime table is UNable? ☺ ☼
to find next prime without integer's
prime multiplication decomposition.

index ; update 2017-05-30




<a name=index> Goldbach conjecture 22=11+11=17+5=19+3 
■ explain Six n Prime Table
■ Six n harmony rule 
■ Why gap sequence 2,6,8 not allow? 
■ Even Meet Prime Table 01
■ Even Meet Prime Table 02
■ Even Meet Prime Table 03
■ Even Meet Prime Table 04
■ Even Meet Prime Table 05
■ Even Meet Prime Table 06
■ Even Meet Prime Table 07
■ Even Meet Prime Table 08
■ Future prime column sum list table
■ evenE = primeA+3 = primeB+primeC 
■ evenF = primeD+3 = primeE+primeF 
■ prime columns: top red, middle blue, bottom black
■ 6*n-1 type prime, (prime+2)+3 all 6*n-1 type
■ 23 to 29, gap=6. sumFogPr < sumFogNP < sumGiven
■ 29 to 31, gap=2. sumFogPr > sumGiven
■ 37 to 41, gap=4. sumFogPr < sumGiven
■ Summary sumFogPr , sumFogNP , sumGiven
■ Even Meet Prime Table and Six N Prime Table emp6npe1.jpg
■ Future prime column sum document 
■ replace string utility, general but no newline 
■ Even Meet Prime Table 09


below 22=11+11=17+5=19+3 ; NOT 22=2*11 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 <a name="docA001"> 2016-08-30-09-25 start On 2016-06-23 upload Prime number and prime decomposition http://freeman2.com/prime_e1.htm prime_e1.htm has graph code and drawing board. On 2016-08-06 upload Prime number calculator http://freeman2.com/prime_e2.htm prime_e2.htm has several prime related calculation buttons and in/output boxes. prime_e2.htm deleted graph code and drawing board to save space. <a name="docA002"> On 2016-09-01 upload Six n Prime Table http://freeman2.com/prim6n01.htm ..... http://freeman2.com/prim6n10.htm On 2016-09-01 upload prime number study notes http://freeman2.com/tute0068.htm After build prime_e1.htm , prime_e2.htm , prim6n01.htm to prim6n10.htm Liu,Hsinhan can study prime number, all study notes record in tute0068.htm <a name="docA003"> First, explain Six n Prime Table http://freeman2.com/prim6n01.htm partial table is next
Six n Prime Table from n=1 to n=20
http://freeman2.com/prime_e2.htm  in 6*n±1  http://freeman2.com/tute0068.htm
6*n-1 prime  number n  6*n+1 prime
5 1 7
11 2 13
17 3 19
23 4 5^2
29 5 31
5^1 * 7^1 6 37
41 7 43
47 8 7^2
53 9 5^1 * 11^1
59 10 61
5^1 * 13^1 11 67
71 12 73
7^1 * 11^1 13 79
83 14 5^1 * 17^1
89 15 7^1 * 13^1
5^1 * 19^1 16 97
101 17 103
107 18 109
113 19 5^1 * 23^1
7^1 * 17^1 20 11^2
2016-08-30-09-40 here 
<a name="docA004">
First prime is 2, second prime is 3. 2*3=6 
Consider 6*n+remainder for n=0 to n=infinity. 
For n=0 
6*n+remainder 0 = 0 
6*n+remainder 1 = 1 
6*n+remainder 2 = 2 
6*n+remainder 3 = 3 
6*n+remainder 4 = 4 
6*n+remainder 5 = 5 
For n=1 
6*n+remainder 0 = 6 
6*n+remainder 1 = 7 
6*n+remainder 2 = 8 
6*n+remainder 3 = 9 
6*n+remainder 4 =10 
6*n+remainder 5 =11 
For n=2 
6*n+remainder 0 =12 
6*n+remainder 1 =13 
6*n+remainder 2 =14 
6*n+remainder 3 =15 
6*n+remainder 4 =16 
6*n+remainder 5 =17 
etc.

<a name="docA005">
For n>0 all 6*n+0, 6*n+2, 6*n+4 are multiple of 
two, not prime. One exception is n=0 case, 
6*0+2=2 is a prime. 
For n>0 all 6*n+0, 6*n+3 are multiple of three. 
One exception is n=0 case, 6*0+3=3 is a prime. 

For general case consideration, drop n=0 special 
case. Start from n=1, we can say 
all 6*n+0, 6*n+2, 6*n+4 are multiple of two.
all 6*n+0, 6*n+3 are multiple of three. 
They are not prime for sure. Only 6*n+1, 6*n+5 
can be prime. 

<a name="docA006">
Twin prime is two consecutive primes differ by 
two. Twin prime is a major concern. Now rewrite 
6*n+5 as 6*n+5+6-6=6*(n+1)+5-6=6*m-1. m=n+1 
We say 6*n+1, 6*n-1 are possibly prime. 

Base on above discussion build Six n Prime Table 
http://freeman2.com/prim6n01.htm
In Six n Prime Table 
No even number, because dropped 6*n, 6*n+2, 6*n+4.
No number multiple of three, because dropped 6*n+3.
Prime2 and prime3 do not fit 6*n±1. Table start 
from prime5,7,11.

<a name="docA007">
Six n Prime Table has three columns. 
Middle column is n number, n used in 6*n±1 
Middle column is called n column or n line. 
Left  column is 6*n-1 numbers. //CPrime 
Right column is 6*n+1 numbers. //BPrime 
These numbers may be prime, may be composite. 
Six n Prime Table list 6*n-1 prime closer to 
n line. List 6*n-1 composite far from n line. 
composite show up as their prime factorization.
Similar consideration for 6*n+1 numbers. 
BPrime CPrime are used in prime_e2.htm Box19 
click button [BPrCPr] which build prime table.

<a name="docA008">
From n_th prime to n+1_th prime, the difference 
is n_th prime gap. g[n]=p[n+1]-p[n]
Please see Six n Prime Table. 
From 6*n-1 prime to 6*n+1 prime, gap is 2.
Example 5+2=7, 5=6*1-1 and 7=6*1+1. 
From 6*n-1 prime to 6*n+1 prime, gap is 8=2+6.
Example 89+8=97, 89=6*15-1 and 97=6*16+1. 
From 6*n-1 prime to 6*n+1 prime, gap is 14=2+12.
Example 113+14=127, 113=6*19-1 and 127=6*21+1. 
From 6*n-1 prime to 6*n+1 prime, gap%6=2 is true
From 6*n-1 prime to 6*n+1 prime, gap%6=4 is false

<a name="docA009">
Above discuss from 6*n-1 prime to 6*n+1 prime.
Below discuss from 6*n+1 prime to 6*m-1 prime.
Please see Six n Prime Table. 
From 6*n+1 prime to 6*m-1 prime, gap is 4.
Example 19+4=23, 19=6*3+1 and 23=6*4-1. 
From 6*n+1 prime to 6*m-1 prime, gap is 10=4+6.
Example 139+10=149, 139=6*23+1 and 149=6*25-1. 
From 6*n+1 prime to 6*m-1 prime, gap is 16=4+12.
Example 1831+16=1847, 1831=6*305+1, 1847=6*308-1. 
From 6*n+1 prime to 6*m-1 prime, gap%6=4 is true
From 6*n+1 prime to 6*m-1 prime, gap%6=2 is false

<a name="docA010">
From 6*n-1 to 6*m+1 , MUST have gap%6=2 
From 6*n+1 to 6*m-1 , MUST have gap%6=4 
This requirement extend to infinity. 
gap%6=0 is from 6*n-1 to 6*m-1, not cross n line.
gap%6=0 is from 6*n+1 to 6*m+1, not cross n line.
Gap cross n line must be in turn. 
If done from 6*n-1 to 6*m+1 , 
next must be from 6*n+1 to 6*m-1 . 
If done from 6*n+1 to 6*m-1 , 
next must be from 6*n-1 to 6*m+1 . 
This is Six n harmony rule, it is 
avoid_Prime3_cut rule.

<a name="docA011">
youtube video [Proving the Riemann hypothesis 3 of 6] 
2:10 say gap [8,6] [6,8] [8,4] [4,8] are allowed  
gap [8,2] [2,8] [8,8] [4,10] [10,4], [10,10] are NOT allowed. 
2:53 say gap [6,6,6] is allowed, but [6,6,6,6] is NOT allowed.
Gap sequence 8,2 ; 10,10 ; 2,6,8 etc 
NEVER SHOW UP, THEY ARE FORBIDDEN. 

<a name="docA012">
Why gap sequence 2,6,8 not allow? Please access  
http://freeman2.com/prime_e2.htm#bx11 
In gap sequence [d] [ input box ] enter 2,6,6 
not enter 2,6,8 because 2,6,8 no match. 
Click  Box11 output match. One 
answer is 149, 151, 157, 163 
Because 2,6,6 add right end 2 get 2,6,8 . You 
can add right end 2 to 163, see why 165 is not 
a prime, why gap sequence 2,6,8 no match. 
Do more experiments, input gap sequence 2,6,10 
see why gap sequence 2,6,8 no match. 
Future add more study notes. 
Liu,Hsinhan 劉鑫漢 2016-08-30-11-40 

<a name="docA013"> update 2017-03-26
2017-03-25-11-07 //please goto tute0067.htm#a603241230
Liu,Hsinhan stopped prime number work about six month. 
2017-03-24-11-?? revisit Goldbach EVen Meet PRIme 1 
http://freeman2.com/gevmpri1.jpg
LiuHH spend half hour find out how to draw gevmpri1.jpg 
Record key steps to tute0067.htm#a603241230 and add 
same notes to http://freeman2.com/gevmpri1.jpg

The reason suddenly revisit gevmpri1.jpg is that 
LiuHH pop up a strange thought, whether gevmpri1.jpg 
contain information allow find future prime without 
do integer's prime factorization. 2200=2^3*5^2*11^1
negative answer is likely. Positive answer is rare. 
2017-03-25-11-22

<a name="docA014"> update 2017-03-30
2017-03-30-11-04 include start
graph
http://freeman2.com/sixnlaw2.jpg
has next document
[[
Prime Gap Six n harmony rule
Prime numbers are those integers which 
is divisible by itself and one. Example
19=19*1 is prime. 20=2*2*5 is not prime
First few primes are 2,3,5,7,11,13,17,
19,23,29,31,37,41,43,47,53,59,61,67....
Formula 6*n-1 and 6*n+1 change n value 
find all primes. See Six n Prime Table
http://freeman2.com/prim6n01.htm 
Left is 6*n-1 prime right is 6*n+1 prime
Define Prime Gap = PrimeNext - PrimeNow 
From 5 to 7 (n=1) from 11 to 13 (n=2) 
the gap is 2=(6*n+1)-(6*n-1)=0+1+1
From 7 to 11, from 13 to 17 need n, n+1
the gap is 4=[6*(n+1)-1]-(6*n+1)=0+6-1-1
Six n Prime Table show that gap 2 cannot 
follow another gap 2. In other words, 
red arrow cannot follow other red arrow
and gap 4 cannot follow another gap 4. 
Blue arrow cannot follow another blue.
gap 2 must follow gap 4, gap 2, gap 4...
Gap sequence 2,2 or 4,4 is not allowed. 
The reason is simple. Explain as next. 
<a name="docA015">
Prime 2,3 do not fit 6*n±1 for any n . 
Start from prime 5,7,11,13,17 ... if p1
and p2 are two neighbor primes p2=p1+2 
Integer sequence p1,p1+2,p1+4 one must 
be divisible by 3. Since p1,p1+2 both 
be prime must be p1=3*m+2 and p2=3*m+4
3*m+2 , 3*m+4 are not divisible by 3.
then p1+4=3*m+6 where 3*m has factor 3, 
6 divisible by 3, then p1+4=3*m+6 has 
factor 3 and p1+4 is not a prime. This 
conclude red arrow cannot follow another
red arrow. Similar reason apply to blue
arrow. Black arrow is gap=6 go downward 
not contribute to "6*n-1 to/from 6*n+1" 
Purple arrow is gap=8 , 8=6+2 in which 
6 no contribution, residual 2 is same as 
red arrow. Primes 139, 149 has gap=10. 
10=6+4 residual 4 is same as blue arrow.
Next line help you to understand 
Six n harmony rule=avoid_Prime3_cut rule
2017-03-29-10-30
<a name="docA016">
Prime gap sequence 2,8 never occur, because 
sequence 2,8 violate Six n harmony rule.
Prime gap sequence 4,16 never occur, because 
in  7, 11, 27 ;  7+4=11, 11+16=27=3*9 
in 13, 17, 33 ; 13+4=17, 17+16=33=3*11 
gap sequence 4,16 third number 27,33 are not
primes. 4,16 is same as 4,16%6 same as 4,4 

http://freeman2.com/prim6n01.htm  Six n Prime Table
http://freeman2.com/prime_e1.htm  Prime drawing board
http://freeman2.com/prime_e2.htm  Prime calculator
http://freeman2.com/tute0067.htm  Goldbach conjecture
http://freeman2.com/tute0068.htm  Prime study notes
http://freeman2.com/sixnlaw2.jpg  this graph URL 
2017-03-29-10-50 Liu,Hsinhan 劉鑫漢 
]]
2017-03-30-11-10 include stop

<a name="docA017"> update 2017-04-08
2017-04-08-16-59 include start
graph
http://freeman2.com/emphtme1.jpg
has next document
[[
Even meet Prime table  http://freeman2.com/prime_e3.htm
even 20=17+3=13+7 , 20 meet prime 3,7,13,17 , not meet 5,11 
x axis: nID=numberID, even=nID*2+6 0,6;1,8;2,10; etc
y axis: pID=primeID, pID=0,1,2,3,... prime=2,3,5,7,...
Blue=6*n-1 prime=5,11,17. Red=6*n+1 prime=7,13,19 etc
Black=prime3. Yellow=even_not_meet prime. Aqua=remote
<a name="docA018">
prime gap show up in three locations 
Any row, yellow is prime gap. No yellow between twinPr
From prime3 go up, each row shift right=prime gap
If draw prime (not pID) row separate=prime gap scale 
In prime_e3.htm below [pID row_align] click get left
with Horizontal on, click [gevmpr00()] output all 
row move to left, show every row are same as prime3
http://freeman2.com/prime_e3.htm utility draw this 
http://freeman2.com/gevmpri0.jpg same graph by VML 
http://freeman2.com/prime_e1.htm utility draw VML 
http://freeman2.com/prime_e2.htm prime calculator
http://freeman2.com/emphtme1.jpg html table graph 
2017-04-07-23-51 Liu,Hsinhan 
劉
鑫
漢
]]

<a name="docA019">
2017-04-08-17-46 start 
Explain "prime gap show up in three locations" 
with graph. See next Even meet Prime table. emphtme1.jpg
Even Meet Prime Table 01
_______________________prime13, upper red row
_______________________prime11, upper blue row
_______________________prime07, lower red row
_______________________prime05, lower blue row
_______________________prime03, prime3 black row
030507__1113__1719__23____2931____37__4143__47prime and Gap
0608101214161820222426283032343638404244464850even number
a604112118 change x-axis to even. <a name="docA020"> prime gap first show See prime3 black yellow row. Left most black square is 6=prime3+prime3. 2nd black square is 8=prime3+prime5. 3rd black square is 10=prime3+prime7. 4th yellow square is 12=prime3+number9. 5th black square is 14=prime3+prime11. 1st black to 2nd black has gap2, no yellow. 3rd black to 5th black has gap4, one yellow. 11th black to 14th black has gap6, two yellow. black yellow row carry prime gap information. Similarly blue yellow row carry prime gap information. red yellow row carry prime gap information. <a name="docA021"> prime gap second show from prime03 black yellow row to prime05 blue yellow row relative to prime03 row, prime05 row shift right gap2 displacement. Similarly from prime07 red yellow row to prime11 blue yellow row relative to prime07 row, prime11 row shift right gap4 displacement. Other row shift follow same pattern. <a name="docA022"> prime gap third show This graph y axis use primeID, not use prime. If y axis use prime, it is easy to see from prime03 row to prime05 row has gap2. from prime07 row to prime11 row has gap4. Hope above explanation help you understand "prime gap show up in three locations" 2017-04-08-18-08 <a name=GoldbachConjecture> update 2017-04-16 2017-04-13-18-36 copied from prime_e3.htm#Goldbach 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 <a name="docA023"> 2017-04-13-18-40 Goldbach conjecture suggest Every even integer (≥6) is the sum of two odd primes. Each prime add prime3 get an even, for example prime03 + prime3 = even06 prime05 + prime3 = even08 prime07 + prime3 = even10 integ09 + prime3 = even12 prime11 + prime3 = even14 Even 6,8,10,14 two prime decomposition involve 3. Even 12 two prime decomposition not involve 3. Because 12=3+9 and 9 is not a prime and fail "sum of two odd primes" Goldbach statement. But 12=5+7 still satisfy Goldbach conjecture. <a name="docA024"> Following is a graph for Goldbach conjecture in small prime range. <a name="docA025"> Liu,Hsinhan choose nID and pID as coordinate units that is because prime 3,5,7,11,13,17,17,23 ... take more space in y axis direction. Same hight less data. primeID 1,2,3,4,5,6,7,8 ... take less space in y axis direction. Same vertical space more data. nID=0,1,2,3,4,5... match even=6,8,10,12,14,16... General relation is even=nID*2+6 pID=0,1,2,3,4,5... match prime=2,3,5,7,11,13,... Function relation is prime=primeArr(pID) <a name="docA026"> 2017-04-13-19-08 here Please see Even Meet Prime Table 02 prime03 + prime3 = even06 is black/yellow row most left black square A. Below A marked 03,06. This 03 is variable prime03 This 06 is even06 (nID=0 not shown) black square A is constant prime3 Even06, variable prime03 and constant prime3 coincide. <a name="docA027"> prime05 + prime3 = even08 is black/yellow row left black square B. Below B marked 05,08. This 05 is variable prime05, it is blue a0 above "B,05,08 column". This 08 is even08 (nID=1 not shown) black square B is constant prime3 Black B(prime3) + blue a0(prime5) = even8. <a name="docA028"> prime07 + prime3 = even10 is "b0,a1,C,07,10" column. This 07 is variable prime07(RED b0) This 10 is even10 (nID=2 not shown) black square C is constant prime3 Black C(prime3) + RED b0 (prime7) = even10. prime5 + prime5 = even10 this is blue a1. First prime05 and second prime5 coincide. <a name="docA029"> integ09 + prime3 = even12 is "b1,a2,D,09,12" column. 3+9=12 this 09 is not a prime then "blackD" become yellowD. Yellow D is prime3. Integer 9(=3*3) is not in this table 12=7+5 both 7 and 5 are primes, 7,5 show up in "b1,a2,D,09,12" column. Blue a2 is prime5, red b1 is prime7. <a name="docA030"> In black/yellow (1)3 row, YELLOW square D alert us that "b1,a2,D,09,12" column not start a new prime row. Because red b1 is second red in "b0,b1,b2,b3,..." row. Red b0 start a new prime row. Red b0 is above BLACK square C, Red b1 not start a new prime row. Red b1 is above YELLOW square D. 2017-04-13-19-55 stop <a name="docA031"> 2017-04-13-22-19 start 14=11+3 is blue c0 + black E 14=7+7 is red b2 + red b2 16=13+3 is red d0 + black F 16=11+5 is blue c1 + blue a4 18=13+5 is red d1 + blue a5 18=11+7 is blue c2 + red b4 20=17+3 is blue e0 + black H 20=13+7 is red d2 + red b5 etc. 2017-04-13-22-26 stop <a name="docA032"> 2017-04-14-15-40 start Goldbach conjecture suggest that even = primeA + primeB ---eq.a01 When we move along x-axis, one variable must change value. In eq.a01 left side even change. In eq.a01 right side two object primeA and primeB. We can set primeB as constant and let primeA change. In black/yellow row primeB=3=constant. In aRow primeB= 5=constant. Even, primeA change. In bRow primeB= 7=constant. same as above. In cRow primeB=11=constant. same as above. In dRow primeB=13=constant. etc. <a name="docA033"> Next explain why black/yellow row all black square form a prime pattern. BlackA: 6=3+3; BlackA=3, BlackA=3 BlackB: 8=5+3; blue a0=5, BlackB=3 BlackC: 10=7+3; red b0=7, BlackC=3 BlackC: 10=5+5; blue a1=5, blue a1=5 YellowD:12=7+5; red b1=7, blue a2=5 YellowD:12=9+3; YellowD=3, 9 is not prime <a name="docA034"> BlackE: 14=11+3; blue c0=11, BlackE=3 BlackE: 14=7+7; red b2=7, red b2=7 BlackF: 16=13+3; red d0=13, BlackF=3 BlackF: 16=11+5; blue a4=5, blue c1=11 YellowG:18=13+5; red d1=13, blue a5=5 YellowG:18=11+7; blue c2=11, red b4=7 YellowG:18=15+3; YellowG=3, 15 is not prime <a name="docA035"> BlackH: 20=17+3; blue e0=17, BlackH=3 BlackH: 20=13+7; red d2=13, red b5=7 BlackI: 22=19+3; red f0=19, BlackI=3 BlackI: 22=17+5; blue e0=17, blue a7=5 BlackI: 22=11+11; blue c4=11, blue c4=11 YellowJ:24=19+5; red f1=19, blue a8=5 YellowJ:24=17+7; blue e2=17, red b7=7 YellowJ:24=13+11; red d4=13, blue c5=11 YellowJ:24=21+3; 21 is not prime <a name="docA036"> BlackK: 26=23+3; blue g0=23, BlackK=3 BlackK: 26=19+7; red f2=19, red b8=7 BlackK: 26=13+13; red d5=13, red d5=13 YellowL:28=23+5; blue g1=23, blue aa=5 YellowL:28=17+11; blue e4=17, blue c7=11 YellowL:28=25+3; 25 is not prime YellowM:30=23+7; blue g2=23, red ba=7 YellowM:30=19+11; red f4=19, blue c8=11 YellowM:30=17+13; blue e5=17, red d7=13 YellowM:30=27+3; 27 is not prime 2017-04-14-16-48 here <a name="docA037"> Please pay attention to above list. All gray background-color primes are even=prime3+NONE_PRIME This NONE_PRIME not match Goldbach conjecture. Thay are yellow square in black/yellow row and out of consideration. Next please pay attention to primes-on-green background-color. Each primes-on-green has a Black#=3. Thay are black squares in Even Meet Prime Table 02. <a name="docA038"> These prime3 has two roles. View horizontally, they are all prime3. View vertically, these black square prime3 match primes 3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61 ... respectively. Because yellow squares represent those odd numbers not match Goldbach conjecture. Above explain black/yellow row, black square has prime number sequential order. How about aRow or "(2)5" row? <a name="docA039"> black/yellow row start from even=6=prime3+prime3 constant is prime3 but in aRow constant is prime5 if in aRow start from even=6=integer1+prime5 integer1 is not a prime, integer1 violate Goldbach conjecture. <a name="docA040"> in aRow start from even=8 =prime3+prime5 next is even=10=prime5+prime5 next is even=12=prime7+prime5 next is even=14=integ9+prime5 next is even=16=prim11+prime5 get same prime number sequential order pattern, only change is constant be prime5 and start from even=8. <a name="docA041"> Similar reason apply to in bRow constant be prime7 and start from even=10. in cRow constant be prime11 and start from even=14. in dRow constant be prime13 and start from even=16. etc. Because prime number has only one sequential, then all rows have identical pattern. In http://freeman2.com/prime_e3.htm#evenMeetPr under "pID rowAlign" change from "normal" to "left" then click RUN ☞ [gevmpr00()] and click DRAW ☞ [show Box22 table] output graph support that in Even Meet Prime Table 02 all rows get prime number sequential order pattern. 2017-04-14-17-33 <a name="docA042"> 2017-04-14-21-08 Even Meet Prime Table 02 is Goldbach conjecture graph. Even Meet Prime Table 03 parallel shift Table 02 all rows to left end. Key point is to see that Table 02 all rows have same structure. Number below black square are primes. Number below yellow columns are non-primes. <a name="docA042"> 2017-04-14-21-15 Next discuss in Even Meet Prime Table 02 each row right shift prime gap distance relative to the row below it. for example jRow j0 j1 j2 j3 j4 j5 (11)37 right shift 6 relative to iRow i0 i1 i2 i3 i4 i5 i6 i7 i8 (10)31 <a name="docA043"> Goldbach conjecture say Every even integer ( ≥ 6 ) is the sum of two odd primes even = primeA + primeB ---eq.a02 jRow is prime37 row. Let primeB=37 get even = primeA + 37 ---eq.a03 here even and primeA are both variables. Smallest primeA possible is primeA=3. Smallest even possible is even = 3 + 37 = 40 ---eq.a04 Assume (wrong) jRow not right shift 6 relative to iRow Assume (wrong) jRow right shift 4 relative to iRow. Then j0 is above 38, even 38 = 1 + 37 in this relation number 1 is NOT a prime and "38 = 1 + 37" violate Goldbach conjecture. <a name="docA044"> In Even Meet Prime Table prime 3 row start even=3+3=6 prime 5 row start even=5+3=8 3 to 5 gap=5-3=2, then start even 8-6 gap=2 ... prime 37 row start even=37+3=40 prime 41 row start even=41+3=44 37 to 41 gap=41-37=4, then start even 44-40 gap=4 ... prime503 row start even=503+3=506 prime509 row start even=509+3=512 509 to 503 gap=509-503=6, start even 512-506 gap=6 ... This relation let each row right shift prime gap distance relative to the row below it. 2017-04-14-21-38 <a name="docA045"> 2017-04-15-16-43 Even Meet Prime Table 02 has prime columns (black square A,B,B,E,F etc.) and has none prime columns (yellow square D,G,J,L,M etc.) Even Meet Prime Table 04 has prime columns and delete none prime columns. <a name="docA046"> Prime3 column is black A one square. Prime5 column is black B and blue a0. Prime7 column is black C and blue a1, red b0. Prime11 column = black E, yellow a3, red b2 and blue c0. Prime13 column = black F, blue a4, yellow b3, blue c1 and red d0. etc. <a name="docA047"> Start from prime5 up to infinity prime, each prime is either type 6*n-1 or type 6*n+1 . Prime3 do not fit 6*n±1, Prime3 column has just one black square A. Prime5 = 6*1-1, square a0 is blue indicate 5 = type 6*n-1 prime. Prime7 = 6*1+1, square b0 is red indicate 7 = type 6*n+1 prime. Prime11= 6*2-1, square c0 is blue indicate 11= type 6*n-1 prime. Prime13= 6*2+1, square d0 is red indicate 13= type 6*n+1 prime. etc. <a name="docA048"> Please study Even Meet Prime Table 04, look like each prime column top square color match that prime 6*n±1 type, but squares below top and above bottom black square all has opposite color from that prime 6*n±1 type. For example Prime19 has top red f0 square and middle e1, c4, a7 blue squares and botton black I. (yellow d3, b6 both do not satisfy Goldbach conjecture, 22=13+9 and 9 is not a prime) Second example Prime23 has top blue g0 square and middle f2, d5, b8 red squares and botton black K. (yellow e3, c6, a9 all do not satisfy Goldbach conjecture, 26=5+21 and 21 is not a prime) Middle squares color opposite to top square color? Is this accidental or is this a must? 2017-04-15-17-30 <a name="docA049"> 2017-04-15-21-00 The following show that in evenE = primeA + 3 ---eq.a05 if primeA is 6*n-1 type (blue) then other smaller two primes sum to evenE must be 6*n+1 type (red). Rewrite primeA + 3 as evenE = 6*n-1 + 3 = 6*n+2 = 6*(q+r)+2 = (6*q+1) + (6*r+1) ---eq.a06 In primeA + 3 = primeB + primeC ---eq.a07 primeA is 6*n-1 type and primeB, primeC both be 6*n+1 type. Example 29+3=19+13 29=6*5-1 (blue) ; 19=6*3+1 (red) ; 13=6*2+1 (red) <a name="docA050"> Next show that in evenF = primeD + 3 ---eq.a08 if primeD is 6*n+1 type (red) then other smaller two primes sum to evenF must be 6*n-1 type (blue). Rewrite primeD + 3 as evenF = 6*n+1 + 3 = 6*n+4 = 6*n+(6-6)+4 = 6*(n+1)-6+4 = 6*(n+1)-2 = (6*q-1) + (6*r-1) ---eq.a09 In primeD + 3 = primeE + primeF ---eq.a10 primeD is 6*n+1 type and primeE, primeF both be 6*n-1 type. Example 37+3=29+11=23+17 37=6*6+1 (red) 29=6*5-1 (blue) ; 11=6*2-1 (blue) ; 23=6*4-1 (blue) ; 17=6*3-1 (blue) ; 2017-04-15-21-25

Box31 input ;


Box32 output ;


Box33 debug ;

QDboxc33.value='' ;
<a name=EMP_to_Prime>
Even meet Prime table is able? to find next prime 
without integer's prime multiplication decomposition.

2017-04-16-05-28 start build Even Meet Prime Table 05
2017-04-16-06-55 done  build Even Meet Prime Table 05



<a name=docA051>
2017-04-16-07-36
Please read above Even Meet Prime Table 05. 
Purple line is a border line. Left to purple 
are given primes. Right to purple is future 
unknown. Task is from primes we have in 
hand find next prime without integer's prime 
multiplication decomposition.

Although future is unknown, but Even Meet Prime 
Table allow us know partial future pattern. 
Right to purple line has fog covered data, they 
are known. Because Even Meet Prime Table all 
rows have identical prime sequence. The only 
difference is each row shift to right a current 
gap distance. 

<a name=docA052>
Next important property help us is that all 
prime column has 
Bottom black square, top blue square, middle 
all be red square. OR
Bottom black square, top red square, middle 
all be blue square. 
To find future prime, we can not see future 
top square be red or blue. But we can see 
future middle squares. If future middle has 
red and blue, mixed color tell us this column 
is not a prime column, stop and exam next gap. 
If future column has all one color, then 
Six N Harmony rule determine this one color 
column is prime or not. Please see first column 
right to purple line. It is m1 to am column. 
m1 to am column has all blue color. If m1 to am 
column is future prime, it has gap 
2017-04-16-07-58 think 
2017-04-16-08-02 I need more rules to determine 
next prime. 

<a name=docA053>
2017-04-16-18-08
Liu,Hsinhan build Even Meet Prime Table 05 and 
hope to predict next prime without integer's 
prime multiplication decomposition. This attempt 
partially achieved, that is 

In Even Meet Prime Table all rows have same 
pattern, all be prime sequence, but shift to 
right at current gap distance. This property 
give us partial future data, see Table 05 
fog covered right half blue-red-yellow squares.

<a name=docA054>
In Even Meet Prime Table all prime columns have 
top square match 6*n±1 property to that prime 
and below top all squares have opposite 6*n±1 
property.

Determine next prime gap, must refer to current 
prime gap in hand. For example, if know from 
2,3,5,7,11 ... 41,43,47 need predict next gap 
current prime gap in hand is 47-43=4, then 
next prime gap size must NOT have nextGap%6=4
This property is governed by Six n harmony rule 
which is avoid_Prime3_cut rule. 

<a name=docA055>
Liu,Hsinhan stopped at Even Meet Prime Table 05 
next gap size determination step. In Table 05 
future m1 to z0 column has all blue, but 49 is 
not a prime. Future m3 to z2 column has all red. 
m3 to z2 predict next prime 53=47+6 Question is 
How to drop m1-z0 column in gap size determination 
step? 

<a name=docA056>
Another example not covered in Table 05 
Assume prime 2,3,5,7 ... 113 are in hand 
next gap is unknown 
from prime 113 to prime 127
prime, gap, gap%6, 6*n±1
 109 ,  4 ,    4 , 6*n+1 <==all known 
 113 , 14 ,    2 , 6*n-1 <==only 113 known, 14 is big target
 127 ,  4 ,    4 , 6*n+1 <==all unknown 
(not create such table, see freeman2.com/gevmpri0.jpg)
Even Meet Prime Table future fog columns are 
Data in hand is 109+gap4=113 
next (unknown) gap must be gap%6=2
116=113+3  <== 113 known
<a name=docA057>
Next is five all same color even numbers between 
prime 113 to prime 127 
118=115+3  <== gap=2%6=2 allowed but not prime
122=119+3  <== gap=6%6=0 allowed but not prime
124=121+3  <== gap=8%6=2 allowed but not prime
128=125+3  <== gap=12%6=0 allowed but not prime
130=127+3  <== gap=14%6=2 allowed 127 is prime
see http://freeman2.com/gevmpri0.jpg
2017-04-16-17-05 How to choose 130=127+3 from 
five same color columns? 
What new rules can be used to 
drop 118=115+3 
drop 122=119+3
drop 124=121+3 
drop 128=125+3 

<a name=docA058>
2017-04-16-18-03 Liu,Hsinhan decide ask general 
public to think this problem. 
Many works should be faster than one work.
2017-04-16-18-39

<a name=docA059> update 2017-04-20
2017-04-19-16-10
In Even Meet Prime Table and given data, all 
prime columns have either 
top red, middle blue, bottom black 
or 
top blue, middle red, bottom black 

Examples are 
red, blue, black 7,13,19,31,37,43 etc.
red, blue, black is for 6*n+1 type primes.
Other examples are 
blue, red, black 5,11,17,29,41,47 etc.
blue, red, black is for 6*n-1 type primes.
See Table 05 left half.  
<a name=docA060>

<a name=docA061>
In Even Meet Prime Table and FUTURE data, 
all prime columns top square are unknown,
all prime columns bottom square are unknown,
FUTURE data prime columns has only middle color. 
Example, Table 05 right half fog area, square 
m3 down to z2 is future prime 53. We do not 
know 53 is prime, then above square m3, we 
cannot draw a blue square. Below square cL 
we cannot draw a black square. 

Compare with none-prime columns, none-prime 
columns have either whole column red,
or     whole column blue,
or     whole column blue AND red.
column contain blue AND red are rejected 
quickly in finding future prime process. But 
whole column red  none-prime columns and
whole column blue none-prime columns 
give us trouble.

<a name=docA062>
In future fog blue,red,yellow area, these 
prime columns and none-prime columns look 
the same, whole column one color, but 
some is prime column, some not.
Even Meet Prime Table 06 show none-prime 
columns. 



<a name=docA063>
Let graph upper left corner be called "sea"
then prime columns and none-prime columns 
has one point different. 
prime columns top square TWO sides face sea,
none-prime columns top square ONE side face  
sea. This is obvious in given data, in future 
fog area, next future row is unknown, we cannot 
create future row. Fog m1 to mm squares, which 
square has one side face sea, which square is 
away from sea that is unclear. 

<a name=docA064>
In Even Meet Prime Table 06 we have all given 
data. Study given data help us work in future 
area. Observe Table 06 
prime 23 to 29 (gap 6), row 25(28) is all blue.
prime 31 to 37 (gap 6), row 35(38) is all red.
prime 47 to 53 (gap 6), row 49(52) is all blue.
prime 53 to 59 (gap 6), row 55(58) is all blue.
prime 61 to 67 (gap 6), row 65(68) is all red.
prime 73 to 79 (gap 6), row 77(80) is all red.
prime 83 to 89 (gap 6), row 85(88) is all blue.
prime 89 to 97 (gap 8), row 91(94) is all blue.
prime 89 to 97 (gap 8), row 95(98) is all red.

<a name=docA065>
Observe 01: all none prime integers 25,49, 
55,85,91 have one color blue.
In primeM+6=primeN these blue columns are 
primeM+2 they are 25,49,55,85
In primeM+8=primeN these blue columns are 
primeM+2 they are 91 (and future not shown)

<a name=docA066>
Observe 02: all none prime integers 35,65,
77,95 have one color red.
In primeM+6=primeN these red columns are 
primeN-2 they are 35,65,77
In primeM+8=primeN these red columns are 
primeN-2 they are 95 (and future not shown)

<a name=docA067>
Observe 03: all none prime integers 9,15,21, 
27,33,39,45,51,57,63,69,75,81,87,93 have TWO 
color red AND blue in one column. These odd 
numbers are multiple of 3 and gap 6.
They are red AND blue in one column, easy to 
identify and not a concern.

<a name=docA068>
2017-04-19-17-21 here 
primeM+6=primeN, integer primeN-2 is all red.
primeM+8=primeN, integer primeN-2 is all red.
How about primeM+10=primeN, whether 
integer primeN-2 is all red? 
prime 139, 149; primeN-2=147, even=150
prime 181, 191; primeN-2=189, even=192
prime 241, 251; primeN-2=249, even=252
prime 283, 293; primeN-2=291, even=294
prime 337, 347; primeN-2=345, even=348

<a name=docA069>
150=139+11 ; 139=6*23+1 red; 11=6*2-1 blue
150=137+13 
150=131+19
150=127+23
150=113+37
150=109+41
150=107+43
150=103+47
150=97+53
150=89+61
150=83+67
150=79+71

note: 150-3=147=3*7*7 , 147 is not a prime 
get ill 150=139+11 = red+blue
2017-04-19-17-35

<a name=docA070>
2017-04-19-19-03 
Next calculate verify 
[[
Observe 01: all none prime integers 25,49, 
55,85,91 have one color blue.
In primeM+6=primeN these blue columns are 
primeM+2 they are 25,49,55,85
]]
<a name=docA071>
Assume primeM is 6*n-1 type prime 
Assume 
primeM+6=primeN ---eq.a11
oM2=primeM+2 ---eq.a12  //odd  M 2 
eM2=oM2+3    ---eq.a13  //even M 2 
For example primeM=23, primeN=27
oM2=23+2=25  //study odd number 25 
eM2=25+3=28  //emp table odd+3=even 
28=23+5      //23=6*4-1,  5=6*1-1
28=17+11     //17=6*3-1, 11=6*2-1

<a name=docA072>
Assume 
eM2=primeS+primeT  ---eq.a14
Hope find that both primeS and primeT are 
6*n-1 type primes. 
From eq.a13 get eM2=oM2+3 , 
apply eq.a14 replace eM2 .
primeS+primeT=oM2+3 , then 
oM2=primeS+primeT-3 ---eq.a15
From eq.a12 get 
oM2=primeM+2  ---eq.a16
<a name=docA073>
Both eq.a15 and eq.a16 are oM2, get 
primeS+primeT-3=primeM+2  ---eq.a17
then 
primeS+primeT=primeM+5  ---eq.a18

Because primeM is 6*n-1 type prime 
//if primeM is 6*n+1 type prime, void following.
primeS+primeT=(6*n-1)+5 =6*n+4=6*(n+1)-2 ---eq.a19
Let integer n+1 be p+q
primeS+primeT=6*(p+q)-2=(6*p-1)+(6*q-1) ---eq.a20

<a name=docA074>
"6*n-1 type prime, (prime+2)+3 all 6*n-1 type" example 
In Even Meet Prime Table 02, box (g0)23+2=25(g1) 
even=25+3=28:23,17,11,5 "23,17,11,5" are all 6*n-1 type
In Even Meet Prime Table 02, box (h0)29+2=31(h1) 
even=31+3=34:29,23,17,11,5 "29,23,17,11,5" are all 6*n-1.
eq.a20 say 
if primeM is blue 6*n-1 type prime , then next 
column odd primeM+2 change to even primeM+5 
primeM+5 must be a sum of two 6*n-1 type 
primes primeS+primeT. Here gapM in 
primeM+gapM=primeN did NOT enter calculation. 
Both next column=prime column satisfy or 
next column=none prime column satisfy.
In table 02 prime 23 row (g0,g1,g2,...gc) is 
blue 6*n-1 type. Square g0 sit on 23 row and 
23 column. Next column g1,e4,c7,aa,L belong 
to above calculation and NOT a prime column. 
column g1,e4,c7,aa,L has all 6*n-1 (blue) 
Squares. On top of g1 is blue sea. (on top 
of g1 is not a red square)
Another example, consider prime 29. 
In table 02 prime 29 row (h0,h1,h2,...h9) is 
blue 6*n-1 type. Square h0 sit on 29 row and 
29 column. Next column h1,g4,e7,ca,ad,O belong 
to above calculation and IS a prime column. 
On top of h1 is i0, i0 is new prime 31 after 
29. (on top of h1 is not blue sea)

2017-04-19-19-32

<a name=docA075>
What happen if primeM is 6*n+1 type prime? 
eq.a19 change to next 
primeS+primeT=(6*n+1)+5 =6*n+6=6*(n+1)+0 ---eq.a21
Let integer n+1 be p+q
primeS+primeT=6*(p+q)+0=(6*p+1)+(6*q-1) ---eq.a22
eq.a22 say 
if primeM is red 6*n+1 type prime , then next 
column odd primeM+2 change to even primeM+5 
primeM+5 must be a sum of one 6*n+1 type 
and one 6*n-1 type primes. Table 02 column 
i1,h2,g5,...ae,P belong to above calculation.
Sum of one 6*n+1 type and one 6*n-1 type 
that is one red add one blue! Not a prime 
column for sure. prime column is 
top red, middle blue bottom black (6*n+1)
top blue, middle red bottom black (6*n-1)
2017-04-19-19-57

<a name=docA076>
2017-04-20-00-07
In Table 05 consider three columns 
m0 L2 k3 ... W greatest known prime 47 
m1 L3 k4 ... z0 unknown zone first column.
m3 L5 k6 ... z2 unknown zone first prime 53.
After 47 we should find 53, but m1 ... z0 
column stand on the way to confuse our 
searching next prime. The following try 
sum all primes in one column and compare 
three sums hope be able drop the confusing 
none prime columns. 

Assume prime 3,5,7,...23 are given. 
Refer to Table 02 square labels, Let 
sumGiven=g0+f2+d5+b8 //drop square K
sumFogNP=g1+e4+c7+aa //drop nothing
sumFogPr=f5+d8    //drop square h0,N
Why drop K(prime3), Why drop h0,N ?
Because in fog future h0,N are unknown.
Since drop N (prime3) then drop K (3)

<a name=docA077>
Sum is next. 
prime 23+3=26 //even 26 = primeA+primeB
26=23+3  //greatest given prime 23 
26=19+7
26=13+13
23+3+19+7+13+13=78 //drop 3(K), since 3(N) unknown
sumGiven=23+19+7+13+13=75 //take 75, ignore 78 

//after given 23, next future uniform color column
28=23+5  //25 is none prime neighbor 
28=17+11 //sumFogNP=sum fog none prime 
sumFogNP=23+5+17+11=56 //drop nothing

//after given 23, next future prime 29 
prime 29+3=32 //even 32 = primeA+primeB 
32=29+3  //drop square h0,N; 
32=19+13
sumFogPr=f5+d8=19+13=32 //drop 29+3
future prime 29 number 29 and 3 are unknown

<a name=docA078>
sumGiven=75 //prime 23
sumFogNP=56 < 75
sumFogPr=32 < 56 //prime 29
Compare data get 
sumFogPr < sumFogNP < sumGiven
prime 23 to prime 29, gap=6. 
Is this a rule for gap=6 ? 

<a name=docA079>
Assume prime 3,5,7,...29 are given. 
Future 31 is unknown. 
sumGiven=column sum of given greatest prime
sumFogNP=column sum of future none prime 
sumFogPr=column sum of future prime 

prime 29+3=32 //even 32 = primeA+primeB
32=29+3  //greatest given prime 29 
32=19+13
sumGiven=29+19+13=61 //prime 29, drop 3

<a name=docA080>
34=31+3  //prime neighbor 
34=29+5
34=23+11
34=17+17
sumFogPr=29+5+23+11+17+17=102 //drop 31+3
sumFogPr=102 > 61=sumGiven
prime 29 to 31, gap=2, sumFogNP not exist 
sumFogPr > sumGiven 
Is this a rule for gap=2 ? 
Is this a rule for gap%6=2 ? 
2017-04-20-00-17

<a name=docA081>
2017-04-20-00-47
Assume prime 3,5,7,...41 are given. 
Refer to Table 02 square labels, Let 
sumGiven=k0+j2+i5+de+bh //drop square T
sumFogPr=k1+h7+ga+ed+aj //drop square L0,U
Because in fog future L0,U are unknown.
Sum is next 

prime 41+3=44 //even 44 = primeA+primeB
44=41+3  //greatest given prime 41 
44=37+7
44=31+13
41+3+37+7+31+13=132 //drop 3
sumGiven=41+37+7+31+13=129

<a name=docA082>
Assume 43 is future prime.
46=43+3  //prime neighbor 
46=41+5
46=29+17
46=23+23
43+3+41+5+29+17+23+23=184 //drop 43+3
sumFogPr=41+5+29+17+23+23=138 > 129
sumFogPr=138 > 129=sumGiven
<a name=docA083>
prime 41 to 43, gap=2, sumFogNP not exist 
sumFogPr > sumGiven 
is this general for gap=2?

<a name=docA084>
Assume prime 3,5,7,...47 are given. 
Integer 49 is none prime neighbor.
Prime 53 is future prime neighbor.

prime 47+3=50 //even 50=primeA+primeB
50=47+3  //greatest given prime 47 
50=43+7
50=37+13
50=31+19
sumGiven=47+43+7+37+13+31+19=197 //drop 3

<a name=docA085>
52=47+5  //none prime neighbor
52=41+11
52=29+23
sumFogNP=47+5+41+11+29+23=156 //drop nothing
sumFogNP=156<197=sumGiven

56=53+3  //next future prime neighbor
56=43+13
56=37+19
sumFogPr=43+13+37+19=112 //dropped 53+3
sumFogPr=112<197=sumGiven
<a name=docA086>
Compare above relation, get 
sumFogPr < sumFogNP < sumGiven
prime 47 to prime 53, gap=6
Is this a rule for gap=6 ?
Is this a rule for gap%6=0 ?
2017-04-20-01-10 done notes 
2017-04-20-07-59 done correction 

<a name=docA087>
2017-04-20-09-37
Above considered 
gap=2 examples
29 to 31 sumFogPr > sumGiven
41 to 43 sumFogPr > sumGiven
gap=6 examples
23 to 29 sumFogPr < sumFogNP < sumGiven
47 to 53 sumFogPr < sumFogNP < sumGiven
The following consider gap=4 example. 

<a name=docA088>
Assume prime 3,5,7,...37 are given. 
Integer 39 is none prime neighbor.
Prime 41 is future prime neighbor.

prime 37+3=40 //even 40=primeA+primeB
40=37+3  //greatest given prime 37 
40=29+11
40=23+17
sumGiven=37+29+11+23+17=117 //drop 3

<a name=docA089>
Integer 39 is none prime neighbor. But 
39 contain factor 3, skip 39.

41 is next future prime
even 44=41+3 //even 44=primeA+primeB
44=41+3  //next future prime 41 is unknown
44=37+7
44=31+13
sumFogPr=37+7+31+13=88 //dropped 41+3
sumFogPr=88<117=sumGiven
<a name=docA090>
prime 37 to 41, gap=4, sumFogNP not exist 
sumFogPr < sumGiven 
is this general for gap=4?

<a name=docA091>
Assume prime 3,5,7,...43 are given. 
Integer 45 is none prime neighbor.
Prime 47 is future prime neighbor.

prime 43+3=46 //even 46=primeA+primeB
46=43+3  //greatest given prime 43
46=41+5
46=29+17
46=23+23
sumGiven=43+41+5+29+17+23+23=181 //drop 3

<a name=docA092>
Integer 45 is none prime neighbor. But 
45 contain factor 3,5, skip 45.

47 is next future prime
even 50=47+3 //even 50=primeA+primeB
50=47+3  //next future prime 47 is unknown
50=43+7
50=37+13
50=31+19
sumFogPr=43+7+37+13+31+19=150 //dropped 47+3
sumFogPr=150<181=sumGiven
<a name=docA093>
prime 43 to 47, gap=4, sumFogNP not exist 
sumFogPr < sumGiven 
is this general for gap=4?

<a name=docA094>
Summary 
gap=2 examples
29 to 31 sumFogPr > sumGiven
41 to 43 sumFogPr > sumGiven
gap=4 examples
37 to 41 sumFogPr < sumGiven
43 to 47 sumFogPr < sumGiven
gap=6 examples
23 to 29 sumFogPr < sumFogNP < sumGiven
47 to 53 sumFogPr < sumFogNP < sumGiven
2017-04-20-10-24

<a name=docA095>
2017-04-20-14-42
Gap 2 primes, 11,13; 17,19; 29,31; 41,43
Gap 4 primes,  7,11; 13,17; 19,23; 37,41; 43,47 
Gap 6 primes, 23,29; 31,37; 
each has special structure. Please open 
http://freeman2.com/prim6n01.htm
and compare prim6n01.htm table top with 
Even Meet Prime Table 02 below.

<a name=docA096>
In prim6n01.htm left column is 6*n-1 type prime.
In Table 02 blue color squares are 6*n-1 type. 

In prim6n01.htm right column is 6*n+1 type prime.
In Table 02 red color squares are 6*n+1 type. 

In prim6n01.htm from left to right is from 6*n-1 
   to 6*n+1 type.
In Table 02 blue blue red is from 6*n-1 to 6*n+1.
   see square label c0,c1,d0; e0,e1,f0; h0,h1,i0; 
see emp6npe1.jpg graph document. 

In prim6n01.htm from right to left is from 6*n+1 
   to 6*n-1 type.
In Table 02 red red red blue is from 6*n+1 to 6*n-1.
   see square b0,b1,b2,c0; d0,d1,d2,e0; f0,f1,f2,g0; 

<a name=docA097>
In prim6n01.htm from left upper to left lower is 
   from 6*n-1 to 6*n-1 type. (type constant)
In Table 02 blue blue blue yellow red is from 6*n-1 
   to 6*n-1. see square g0,g1,g2,g3,h0; 
   Both g0 and h0 are blue (23,29 both be 6*n-1)
see emp6npe1.jpg graph document. 

In prim6n01.htm from right upper to right lower is 
   from 6*n+1 to 6*n+1 type. (type constant)
In Table 02 red red red yellow blue is from 6*n+1 
   to 6*n+1. see square i0,i1,i2,i3,j0; 
   Both i0 and j0 are red (31,37 both be 6*n+1)
Type constant two primes must have gap%6=0. That 
is two primes must have gap=6,12,18,24 ... each 
gap divide by 6, residual zero.

<a name=docA098>
In prim6n01.htm from left to right is from 6*n-1 
   to 6*n+1 type. Immediately again left to right 
   is impossible, because no seats in right side 
   "outer space" and because cut off by prime3. 
   In 11,13,15 third 15 is not a prime. 
In Table 02 blue blue red is from 6*n-1 to 6*n+1.
   Never immediately again blue blue red.
In Table 02 never see red red up_blue. Because 
   red red up_blue enter "outer space" no seats.
see emp6npe1.jpg graph document. 

<a name=docA099>
In prim6n01.htm from right to left is from 6*n+1 
   to 6*n-1 type. Immediately again right to left 
   is impossible, because no seats in left side 
   "outer space" and because cut off by prime3. 
   In 19,23,27 third 27 is not a prime. 
In Table 02 red red red blue is 6*n+1 to 6*n-1.
   Never immediately again red red red blue.
In Table 02 never see blue blue blue up_red. 
   Since blue blue blue up_red enter "outer space" 
   no seats there.
   Although e0,e1,e2,f1 is blue blue blue up_red,
   but f1 not start a new prime row. Only f0 start
   a new prime row.
see emp6npe1.jpg graph document. 
2017-04-20-15-32

<a name=docA100>
2017-04-20-15-52
Previous update 2017-04-16, in 
Even Meet Prime Table 05 
mark Lj,Lk,LL,LM four continuous fog red.
This is an error, three continuous red or  
three continuous blue, this is impossible. 
Except initial special case, prime 3,5,7 
has gap 2,2 all other primes must not have 
gap 2,2. Four continuous fog red represent 
gap 2,2,2. Even initial special case do not 
have this 2,2,2 !
update 2017-04-20 change square LL to yellow. 
2017-04-20-15-59

<a name=docA101>
2017-04-20-18-08
One hour ago created 
http://freeman2.com/emp6npe1.jpg
Even Meet Prime Table and Six N Prime Table 
comparison graph. List graph below 

2017-04-20-18-10

<a name=docA102> update 2017-04-23
2017-04-22-17-22
Above sum purple vertical line neighbor column 
color square value to sumFogPr, sumFogNP and 
sumGiven. Hope compare sum values to decide 
find next prime number. 
The following is another attempt. Different 
direction but same goal. 
Even Meet Prime Table 07 is same as 
Even Meet Prime Table 05 slight change at 
Table 07 y axis use odd number (Table 05 pID)
Table 07 insert a gray vertical column, top 
mark 47, bottom mark $. Call this $ column.
<a name=docA103>

<a name=docA104>
$ column is copied from given black/yellow row.
Black/yellow row has gaps, $ column follows 
and vertical direction must insert gap rows. 
Compare $ column with m1 to z0 column, 
what reason tell us to reject m1 to z0 column 
(49) as a prime column?
what reason tell us to choose m3 to z2 column 
(53) as next prime column?
Liu,Hsinhan did NOT find out reason, Liu upload 
what is in hand let reader work together.
2017-04-22-17-38 



<a name=empt08>




<a name=empt09>



<a name=ColumnSum31to61>
   Future prime column sum list columnTop constant
column typeodd numbereven numbercolumn sum6*n-1 drop6*n+1 dropbluPrAllredPrAllbluPrSum redPrSumcolumnTop colBottom
prime313411903985708531313
3*n 333614411085707470315
odd 35385780138070057317
prime3740800638063800319
3*n 3942126170806363633111
prime414444691969630443113
prime43466905069506903115
3*n 454896230695046503117
prime47505052052500503119
odd 49525203152315203121
3*n 515454290523123313123
prime5356029312931003125
odd 55582903129312903127
3*n 57606000293129313129
prime596231000310313131
prime616400000003133
First data row is given prime. Following rows are future data. Goal: find next prime. Odd+prime3=even, 35+3=38.
Even Meet Prime analysis 38=31+7=19+19, 38 meet 7,19,31 sum to 57. but 38 not meet 5,7,11,13,17,19,23,29,31.
Even 38 column top=31=given prime, 38 column bottom=7, drop 3,5(<7) because prime 3,5 are not visible.
Primes (3,5,)7,11,13,17,19,23,29,31 < 38-3; Primes >=5 has 6*n-1 and 6*n+1 two types. 11=6*2-1, 19=6*3+1.
6*n-1 primes are 11,17,23,29; bluPrAll=80=11+17+23+29 All four 6*n-1 primes not meet 38, then [6*n-1 drop]=80
6*n+1 primes: 7,13,19,31; redPrAll=70=7+13+19+31; 13(=6*2+1) not meet 38, [6*n+1 drop]=13 (38-13=25 nonPr)
Can you see any reason support 37 be next prime after 31 ? More at http://freeman2.com/tute0068.htm
Liu,Hsinhan 劉鑫漢 build table and document on 2017-04-29-12-00 Table start 2017-04-24-18-50 (a604241850)
<a name=ColumnSumDoc> update 2017-05-03
2017-05-02-15-43
'Column' in 'ColumnSumDoc' mean Columns in 
Even Meet Prime Table 07 and in 
Even Meet Prime Table 08 Especially for columns 
in future fog zone.
'Column' in 'ColumnSumDoc' do NOT mean Columns 
in Future prime column sum list
2017-05-02-15-49

<a name=docA105>
2017-04-29-17-21 start 
In http://freeman2.com/prime_e3.htm#bx24 
added one section 
Future prime column sum list table builder
For Even Meet Prime Table 08 given prime from 5 to 31, 
the column sum list table is here
In column sum list, first row is given prime 31.
column typeodd number even numbercolumn sum 6*n-1 drop6*n+1 drop bluPrAllredPrAll bluPrSumredPrSum columnTopcolBottom
prime31 34119 039 8570 8531 313
Following rows are all future data. The goal is to find whether we can determine next prime 37 without using integer's prime multiplication decomposition. <a name=docA106> Column sum list table structure is the following. The main column is blue/red/silver color column. column type tell current odd number is a prime? or a 3*n number or just an odd number (like 49)
column type
prime
3*n
odd
Odd number column start from user assigned prime up to user assigned end number. Each row increase 2 to pass even numbers.
odd number
31
33
35
<a name=docA107> Even number column start from user assigned prime add three.
even number
34
36
38
The reason to add 3 (not add 5,7,11 etc.) that is because all prime (example 31) add prime3 become an even number (example 31+3=34) Prime 31 first time show up at even 34=31+3. Prime 31 not first show up at even 36=31+5. Prime 31 not first show up at even 38=31+7. "add 3" become a guide line. Odd 35 also add 3 get 35+3=38 and 38 is used to do Even Meet Prime analysis for odd 35. (35 is not a prime) <a name=docA108> column sum has red/silver/pink color squares.
column sum
119
144
57
80
red color square indicate this sum for a prime. silver color square indicate this sum for a 3*n. pink color square indicate this sum for an odd. Top first red is from given greatest prime 31. 31+3=34, 34=31+3=29+5=23+11=17+17 34 meet 3,5,11,17,23,29,31 3+5+11+17+23+29+31=119 <a name=docA109> Top second red is from next unknown prime. Is it possible that column sum number sequence reveal next unknown prime? silver color square, for example odd 33 even 36, has higher sum value. If odd is not 3*n then Even Meet Prime analysis has all 6*n-1 type prime (55, 58) or sum has all 6*n+1 type prime (77, 80). But if odd is 3*n then both 6*n-1 type and 6*n+1 type prime enter sum. Sum value is higher. 2017-04-29-18-18 <a name=docA110> column sum is the major column, this column sum what value? Even Meet Prime analysis has two methods. First 40=37+3=29+11=23+17 in http://freeman2.com/prime_e3.htm#bx14 Box enter 40 , choose Click output to Box15 40=37+3 40=29+11 40=23+17 <a name=docA111> Second method in http://freeman2.com/prime_e3.htm#bx24 Enter 37 and 41
prime bgn , end RUN 21 ⇒
Click [evenMeetPrim0()] get nID and pID 
17 , 1,4,6,8,9,11
18 , 2,4,5,7,8,9,10,11
19 , 1,3,5,10,11,12
17 is numberID, 17*2+6=even 40 
11 is primeID, primeArr[11]=37
click [ID2Pr()] get even and prime 
40:3,11,17,23,29,37
42:5,11,13,19,23,29,31,37
44:3,7,13,31,37,41

<a name=docA112>
For even 40 (odd 37) the sum is 
3+11+17+23+29+37=120
But Future prime column sum list get 
11+17+23+29=80
Sum list dropped 37, because given prime 31, 
future prime 37 is unknown. 
Sum list dropped 3, because in Table 08 
i3,h4 ... z2,37 column do not have prime 3,5,7
Program must drop 3,5,7. Limited future data 
let even 40 sum 3+11+17+23+29+37=120 cut to 
11+17+23+29=80
In Table 08 go further right future, correspond 
to sum list go further down. In hand data is 
less and less. 
2017-04-29-18-55

<a name=docA113>
2017-05-01-18-17 
On 2017-04-30 Liu,Hsinhan added next checkbox 
Box25 output ;Help ,;unknown future column sum
to http://freeman2.com/prime_e3.htm#run2122
If checkbox is unchecked, yellow banner print 
unknown future column sum
If checkbox is checked, yellow banner print 
ALL GIVEN column sum list

The choice "unknown future column sum" is 
Future prime column sum list table builder 
original goal. Consider future prime is unknown. 
Even Meet Prime Table 05 is major concern. 

<a name=docA114>
The choice "ALL GIVEN column sum list" is new 
added section. Because human already known prime 
up to 10^22. Within the known range, we are free 
to set all data be given. To compare future mode 
answer with ALL GIVEN mode answer it is easy to 
find out the error in future mode. To add "ALL 
GIVEN mode" code, the effort is minimum, just 
modify columnTop and colBottom range. 
columnTop colBottom <= left narrower columnTop colBottom
313 <= future unknown31 3
315 <= main study33 3
317 right wider =>35 3
319 right all given =>37 3
ALL GIVEN mode [columnTop, colBottom] wider and wider future mode [columnTop, colBottom] narrower and narrower 2017-05-01-18-39 <a name=docA115> 2017-05-02-10-54 Next explain Future prime column sum list columnTop constant. prime bgn 31, end 37
6*n-1 drop6*n+1 dropbluPrAllredPrAllbluPrSumredPrSum
03985708531
11085707470
80138070057
0638063800
Prime 31 + 3 = even 34=31+3=29+5=23+11=17+17 34:3,5,11,17,23,29,31 even 34 not meet 7,13,19 34 drop 7+13+19=39. Under [6*n+1 drop] first entry is 39. Odd 33 + 3 = even 36=31+5=29+7=23+13=19+17 36:5,7,13,17,19,23,29,31 even 36 not meet 11 36 drop 11. Under [6*n-1 drop] 2nd entry is 11. <a name=docA116> Wider table is next. (different begin/end) All given prime column sum list colBottom constant prime bgn 5, end 13
column typeodd numbereven numbercolumn sum6*n-1 drop6*n+1 dropbluPrAllredPrAllbluPrSumredPrSumcolumnTopcolBottom
prime58800505053
prime7101500575773
3*n 912121101675793
prime11142150167117113
prime1316320716201613133
In this section, [6*n-1 drop] and [6*n+1 drop] are main concern. Follow Goldbach conjecture, carry out Even number two prime sum decomposition. 14=11+3=7+7 ; 14:3,7,11 Even 14 not meet prime5 and 5=6*1-1 Under [6*n-1 drop] column and [prime 11 14 ] row find entry 5. It say Even number two prime sum decomposition dropped 5. Similarly 16=13+3=11+5 ; 16:3,5,11,13 Even 16 not meet 7 , Under [6*n+1 drop] column and [prime 13 16 ] row find entry 7. 2017-05-02-11-39 <a name=docA117> 2017-05-02-14-24 How to get [6*n-1 drop] and [6*n+1 drop] ? The calculation is under [bluPrAll] sum all 6*n-1 primes and [redPrAll] sum all 6*n+1 primes followed with calculation [bluPrSum] sum meet 6*n-1 primes and [redPrSum] sum meet 6*n+1 primes Finally all_sum subtract meet_sum get drop_sum. Liu,Hsinhan include [6*n-1 drop] and [6*n+1 drop] hope to get more future data and help to find next gap (future, unknown). 2017-05-02-14-41
<a name=docA118> update 2017-05-05
2017-05-05-16-42 merge [All given prime column sum list] with [Future prime column sum list] .
Both table (given and unknown) come from http://freeman2.com/prime_e3.htm reader need cut
and paste merge to one table.
Assume primes from 3,5,7 ... 2549,2551,2557 are given. Future are unknown. Goal is to find
next (after 2557) gap size. If find a way to identify next gap size, then next prime=2557+gap
Do you have any good idea? Liu,Hsinhan 2017-05-05-17-01
All given prime column sum list colBottom constant
column typeodd numbereven numbercolumn sum6*n-1 drop6*n+1 dropbluPrAllredPrAllbluPrSumredPrSumcolumnTopcolBottom
prime125032506115276103091204946215861207449112770250325033
3*n 25052508185592122395115323215861207449934669212625053
odd 2507251011295021586194499215861207449011295025073
odd 250925128540813045320744921586120744985408025093
3*n 25112514181008129125113177215861207449867369427225113
odd 251325169560821586111184121586120744909560825133
odd 251525189190712395420744921586120744991907025153
3*n 25172520282240723756869521586120744914348613875425173
odd 251925228827021586111917921586120744908827025193
prime125212524103484114901207449215861209970100960252125213
3*n 25232526171768130579123484215861209970852828648625233
odd 252525288595221586112401821586120997008595225253
odd 2527253013915076711209970215861209970139150025273
3*n 25292532177240134229114362215861209970816329560825293
prime025312534103894215861108610218392209970253110136025313
odd 253325369129612709620997021839220997091296025333
3*n 25352538177660127130123572218392209970912628639825353
odd 25372540109220218392100750218392209970010922025373
prime1253925429659612433820997021839221250994054253925393
3*n 25412544195888119037115976218392212509993559653325413
prime0254325469165621839212339922093521250925438911025433
odd 2545254812230498631212509220935212509122304025453
3*n 25472550252450905349046022093521250913040112204925473
prime025492552109736220935105325223484212509254910718425493
prime125512554103437122601212509223484215060100883255125513
3*n 25532556181476134473122595223484215060890119246525533
odd 255525587801922348413704122348421506007801925553
prime125572560122880103164215060223484217617120320255725573
3*n 2559256221008412340210761522348421761710008211000225575
3*n 2559256221008412340210761522348421761710008211000225593
odd 256125648717622347913044122347921761708717625577
odd 256125648717622348413044122348421761708717625613
odd 256325669365912982021761022347921761093659025579
odd 256325669365912982521761722348421761793659025633
3*n 256525681643521452071315302234792176107827286080255711
3*n 25652568164352145212131537223484217617782728608025653
odd 256725701130802234681045302234682176100113080255713
odd 25672570113080223484104537223484217617011308025673
odd 256925721003081231602175972234682175971003080255715
odd 25692572100308123176217617223484217617100308025693
3*n 25712574221364111462108239223468217597112006109358255717
3*n 2571257422136411147810825922348421761711200610935825713
odd 257325761107682234511068292234512175970110768255719
odd 25732576110768223484106849223484217617011076825733
odd 2575257891519131932217578223451217578915190255721
odd 257525789151913196521761722348421761791519025753
3*n 2577258024510010139294537223451217578122059123041255723
3*n 257725802451001014259457622348421761712205912304125773
prime02579258289079223428128499223428217578089079255725
prime0257925829166122348412853822606321761725798907925793
odd 258125841085281149002175782234282175781085280255727
3*n 258325861861921292761255382234282175789415292040255729
odd 2585258885404223399132174223399217578085404255731
odd 25872590152810705892175472233992175471528100255733
3*n 258925921736641381221291602233992175478527788387255735
prime02591259486899223399130648223399217547086899255737
column typeodd numbereven numbercolumn sum6*n-1 drop6*n+1 dropbluPrAllredPrAllbluPrSumredPrSumcolumnTopcolBottom
Prime 2503 to 2557 are given. Odd 2559 to 2591 are unknown. Goal is to find next (after 2557) gap size.
This table main purpose to to show you merge two table to one. 2017-05-05-17-05
Output may contain error. Please verify first.
<a name=docA119> update 2017-05-08 
2017-05-06-15-43 
if q is a prime, what condition is necessary 
for (q+2) is also a prime? 
q%3 >0 , (q+2)%3 >0 
q%5 >0 , (q+2)%5 >0 
q%7 >0 , (q+2)%7 >0 
q%11>0 , (q+2)%11>0 
q%13>0 , (q+2)%13>0 
q%17>0 , (q+2)%17>0 
.....


2017-05-06-15-56
59%3 
2
59%5 
4
59%7 
3
59%11
4
59%13
7
59%17
8
59%19
2
59%23
13
59%29
1
59%31
28
59%37
22
59%39
20
59%41
18
59%43
16
59%47
12
59%53
6


2017-05-06-15-57
59%3 +2
4
59%5 +2
6
59%7 +2
5
59%11+2
6
59%13+2
9
59%17+2
10
59%19+2
4
59%23+2
15
59%29+2
3
59%31+2
30
59%37+2
24
59%39+2
22
59%41+2
20
59%43+2
18
59%47+2
14
59%53+2
8

<a name=docA120>
2017-05-06-15-59
59%3 +4
6        6%3=0  59+4=63 is not a prime 
59%5 +4
8
59%7 +4
7        7%7=0  59+4=63=9*7 is not a prime 
59%11+4
8
59%13+4
11
59%17+4
12     12%17=12
59%19+4
6       6%19=6
59%23+4
17
59%29+4
5
59%31+4
32
59%37+4
26
59%39+4
24     24%39=24
59%41+4
22
59%43+4
20
59%47+4
16
59%53+4
10

<a name=docA121>
2017-05-06-16-05 integer's prime multiplication 
decomposition is reliable. 

[[
Even meet Prime table is able? ☺ ☼ 
to find next prime without integer's 
prime multiplication decomposition. 
]]
2017-05-06-16-18 UNable
C:\$fm\upload\fm\tute0068.htm
[[
Even meet Prime table is UNable? ☺ ☼ 
to find next prime without integer's 
prime multiplication decomposition. 
]]
2017-05-06-16-24

<a name=docA122>
2017-05-06-17-17
In All given/Future prime column sum list 
prime0 is 6*n-1 type prime 5,11,17,23 etc.
prime1 is 6*n+1 type prime 7,13,19,31 etc. 
6*n±1 type prime is a major concern in 
Even meet Prime table.
2017-05-06-17-22

<a name=docA123>
2017-05-08-11-45
update 2017-05-08 in All given prime column 
sum list add purple line for given and 
future border line. In future section (below 
purple line) inserted blue lines between 
yellow lines. Blue background lines are from 
all given prime data, yellow background lines 
are from future data. Purple line in sum list 
is same as purple line in Table 08

<a name=docA124>
update 2017-05-08 change from
Even meet Prime table is able? ☺ ☼ to find ...
to 
Even meet Prime table is UNable? ☺ ☼ to find ...
Liu, HsinHan studied even meet Prime table 
several days, did not find a method to use 
future data in hand do useful calculation. 
Add weight to suspicious end. 
2017-05-08-12-01

<a name=docA125> 
2017-05-30-15-09 
update 2017-05-30 
Please see Even Meet Prime Table 05 
From this table each row is a prime sequence, but 
shift to right a prime_gap distance. If exam its 
columns, it has four type different columns.
6*n+1 Prime column, for example 19=6*3+1, 31=6*5+1 
      etc. Column has red hat, blue cloth, black shoe.
6*n-1 Prime column, for example 23=6*4-1, 41=6*7-1 
      etc. Column has blue hat, red cloth, black shoe.
3*n odd column, for example 21=3*7, 45=3*15 etc. 
      column has red square/blue square in turn, 
      column has no black shoe.
5*n 7*n odd column, for example 25=5*5, 49=7*7 etc. 
      column has all red or all blue, no hat, no shoe.

<a name=docA126>
Liu,Hsinhan had a question:
If insert more none-prime rows, will change of row 
alter column characters? In Table 09 added 
[35 no no] row and added [25 no no] row. In 
Even Meet Prime Table all red row are 6*n+1 
primes, all blue row are 6*n-1 primes. 
Because 35=6*6-1 [35 no no] row color blue. 
Because 25=6*4+1 [25 no no] row color red. The 
result is Even Meet Prime Table 09. This table 
show that 
If insert more none-prime rows, this insertion 
will not change column characters.
<a name=docA127>
Study Even Meet Prime Table 07 and Table 09 
after maximum given prime 47, the determination 
of next prime cannot rely on column pattern, 
the determination of next prime 53 can be done 
by integer's prime multiplication decomposition 
reject 49=7*7, reject 51=3*17 . 
Can you find next prime gap from future data 
in hand?
Can you find prime rules from future data 
in hand? 
Liu,Hsinhan 劉鑫漢 2017-05-30-15-51

<a name=docA128>
2017-05-30-17-32 
on 
2017-05-30-16-16 get the idea 
study Even Meet Prime Table 07 column pattern 
see what information it carry. Build 
Even Meet Prime Table 10a 
Even Meet Prime Table 10b
Even Meet Prime Table 10c
Difference is the purple column mark. 

<a name=empt10>


<a name=empt10b>


<a name=empt10c>

2017-05-30-17-35 stop

[=][][] 


<a name="bx34">
Box34 input ; replace string utility, general but no newline. Click [Box34] [replaceAA]

from to ; from to
from to ; from to
Box35 output ; example to , RUN ☞


Box36 debug ;

QDboxc36.value='' ;
 


<a name="docA999">

x0+i*y0 x0+∆x
 
ux=vy
ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡ΢ΣΤΥΦΧΨΩ ┌│┐│ 
ΪΫάέήίΰ 
αβγδεζηθικλμνξοπρςστυφχψω 

<a name="NumberSetsChar">

ℂ Complex numbers ; 複數
ℍ Hello ; 
ℕ Natural numbers ; 自然數(正整數及零)
ℙ Prime numbers ; 素數
ℚ Quotient, Rational numbers ; 有理數
ℝ Real numbers ; 實數
ℤ Zahl, Integers ; (from Zahl, German for integer) ; 
ℤ 整數(正整數及零及負整數)
2015-03-13-18-52







Javascript index
http://freeman2.com/jsindex2.htm   local
Save graph code to same folder as htm files.
http://freeman2.com/jsgraph2.js   local


Prime number study notes
file name tute0068.htm mean
TUTor, English, 68 th .htm

2016-08-08-18-47 save as tute0068.htm


The address of this file is
http://freeman2.com/tute0068.htm
First upload 2016-09-01

Thank you for visiting Freeman's page.
Freeman Liu,Hsinhan 劉鑫漢
2016-08-29-19-32