﻿ Prime number study notes , tute0068 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

2016-08-30-09-25 start
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.

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

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.
Liu,Hsinhan　劉鑫漢 2016-08-30-11-40

<a name="docA013"> update 2017-03-26
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
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.
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.
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

]]

<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.
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.

"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.
t068tbl2()
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.
t068tbl2()
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

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.

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.
t068tbl3()

<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.
t068tbl2()
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

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.
t068tbl4()
<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

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)
t068tbl4()
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

t068tbl2()

```

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

t068tbl5()

<a name=docA051>
2017-04-16-07-36
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.

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>
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.
t068tbl5()
<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.

t068tbl6()

<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.

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.
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

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)
t068tbl2()
2017-04-19-19-32

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

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

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
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

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.
t068tbl2()
<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

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>
t068tbl7()
<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>
t068tbl8()

<a name=empt09>
t068tbl9()
```

<a name=ColumnSum31to61>
Future prime column sum list columnTop constant
 column type odd number even number column sum 6*n-1 drop 6*n+1 drop bluPrAll redPrAll bluPrSum redPrSum columnTop colBottom prime 31 34 119 0 39 85 70 85 31 31 3 3*n 33 36 144 11 0 85 70 74 70 31 5 odd 35 38 57 80 13 80 70 0 57 31 7 prime 37 40 80 0 63 80 63 80 0 31 9 3*n 39 42 126 17 0 80 63 63 63 31 11 prime 41 44 44 69 19 69 63 0 44 31 13 prime 43 46 69 0 50 69 50 69 0 31 15 3*n 45 48 96 23 0 69 50 46 50 31 17 prime 47 50 50 52 0 52 50 0 50 31 19 odd 49 52 52 0 31 52 31 52 0 31 21 3*n 51 54 54 29 0 52 31 23 31 31 23 prime 53 56 0 29 31 29 31 0 0 31 25 odd 55 58 29 0 31 29 31 29 0 31 27 3*n 57 60 60 0 0 29 31 29 31 31 29 prime 59 62 31 0 0 0 31 0 31 31 31 prime 61 64 0 0 0 0 0 0 0 31 33
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

2017-04-29-17-21 start
In http://freeman2.com/prime_e3.htm#bx24
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

even number
34
36
38
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.
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

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
up to 10^22. Within the known range, we are free
to set all data be given. To compare future mode
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

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 type odd number even number column sum 6*n-1 drop 6*n+1 drop bluPrAll redPrAll bluPrSum redPrSum columnTop colBottom prime1 2503 2506 115276 103091 204946 215861 207449 112770 2503 2503 3 3*n 2505 2508 185592 122395 115323 215861 207449 93466 92126 2505 3 odd 2507 2510 112950 215861 94499 215861 207449 0 112950 2507 3 odd 2509 2512 85408 130453 207449 215861 207449 85408 0 2509 3 3*n 2511 2514 181008 129125 113177 215861 207449 86736 94272 2511 3 odd 2513 2516 95608 215861 111841 215861 207449 0 95608 2513 3 odd 2515 2518 91907 123954 207449 215861 207449 91907 0 2515 3 3*n 2517 2520 282240 72375 68695 215861 207449 143486 138754 2517 3 odd 2519 2522 88270 215861 119179 215861 207449 0 88270 2519 3 prime1 2521 2524 103484 114901 207449 215861 209970 100960 2521 2521 3 3*n 2523 2526 171768 130579 123484 215861 209970 85282 86486 2523 3 odd 2525 2528 85952 215861 124018 215861 209970 0 85952 2525 3 odd 2527 2530 139150 76711 209970 215861 209970 139150 0 2527 3 3*n 2529 2532 177240 134229 114362 215861 209970 81632 95608 2529 3 prime0 2531 2534 103894 215861 108610 218392 209970 2531 101360 2531 3 odd 2533 2536 91296 127096 209970 218392 209970 91296 0 2533 3 3*n 2535 2538 177660 127130 123572 218392 209970 91262 86398 2535 3 odd 2537 2540 109220 218392 100750 218392 209970 0 109220 2537 3 prime1 2539 2542 96596 124338 209970 218392 212509 94054 2539 2539 3 3*n 2541 2544 195888 119037 115976 218392 212509 99355 96533 2541 3 prime0 2543 2546 91656 218392 123399 220935 212509 2543 89110 2543 3 odd 2545 2548 122304 98631 212509 220935 212509 122304 0 2545 3 3*n 2547 2550 252450 90534 90460 220935 212509 130401 122049 2547 3 prime0 2549 2552 109736 220935 105325 223484 212509 2549 107184 2549 3 prime1 2551 2554 103437 122601 212509 223484 215060 100883 2551 2551 3 3*n 2553 2556 181476 134473 122595 223484 215060 89011 92465 2553 3 odd 2555 2558 78019 223484 137041 223484 215060 0 78019 2555 3 prime1 2557 2560 122880 103164 215060 223484 217617 120320 2557 2557 3 3*n 2559 2562 210084 123402 107615 223484 217617 100082 110002 2557 5 3*n 2559 2562 210084 123402 107615 223484 217617 100082 110002 2559 3 odd 2561 2564 87176 223479 130441 223479 217617 0 87176 2557 7 odd 2561 2564 87176 223484 130441 223484 217617 0 87176 2561 3 odd 2563 2566 93659 129820 217610 223479 217610 93659 0 2557 9 odd 2563 2566 93659 129825 217617 223484 217617 93659 0 2563 3 3*n 2565 2568 164352 145207 131530 223479 217610 78272 86080 2557 11 3*n 2565 2568 164352 145212 131537 223484 217617 78272 86080 2565 3 odd 2567 2570 113080 223468 104530 223468 217610 0 113080 2557 13 odd 2567 2570 113080 223484 104537 223484 217617 0 113080 2567 3 odd 2569 2572 100308 123160 217597 223468 217597 100308 0 2557 15 odd 2569 2572 100308 123176 217617 223484 217617 100308 0 2569 3 3*n 2571 2574 221364 111462 108239 223468 217597 112006 109358 2557 17 3*n 2571 2574 221364 111478 108259 223484 217617 112006 109358 2571 3 odd 2573 2576 110768 223451 106829 223451 217597 0 110768 2557 19 odd 2573 2576 110768 223484 106849 223484 217617 0 110768 2573 3 odd 2575 2578 91519 131932 217578 223451 217578 91519 0 2557 21 odd 2575 2578 91519 131965 217617 223484 217617 91519 0 2575 3 3*n 2577 2580 245100 101392 94537 223451 217578 122059 123041 2557 23 3*n 2577 2580 245100 101425 94576 223484 217617 122059 123041 2577 3 prime0 2579 2582 89079 223428 128499 223428 217578 0 89079 2557 25 prime0 2579 2582 91661 223484 128538 226063 217617 2579 89079 2579 3 odd 2581 2584 108528 114900 217578 223428 217578 108528 0 2557 27 3*n 2583 2586 186192 129276 125538 223428 217578 94152 92040 2557 29 odd 2585 2588 85404 223399 132174 223399 217578 0 85404 2557 31 odd 2587 2590 152810 70589 217547 223399 217547 152810 0 2557 33 3*n 2589 2592 173664 138122 129160 223399 217547 85277 88387 2557 35 prime0 2591 2594 86899 223399 130648 223399 217547 0 86899 2557 37 column type odd number even number column sum 6*n-1 drop 6*n+1 drop bluPrAll redPrAll bluPrSum redPrSum columnTop colBottom
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

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
[[
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.
2017-05-08-12-01

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>
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>
t068tbl10(0)

<a name=empt10b>
t068tbl10(1)

<a name=empt10c>
t068tbl10(2)
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='' ;
```

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
```

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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
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