﻿ Pay6 small prime gap data study notes, tute0070 Pay6 small prime gap data study notes tute0070
This file tute0070.htm is document for http://freeman2.com/pay6get2.htm
and http://freeman2.com/pay6nxt2.htm
tute0070.htm and pay6get2.htm and pay6nxt2.htm
do not have new concept, but use an uncommon
prime gap notation
; index
 The total count of prime gap pattern [2,2] is one. initial special The total count of prime gap pattern [14,20] is zero. Because 14%6=20%6=2 The total count of prime gap pattern [gapA%6=2,gapB%6=2] is zero. GapRule The total count of prime gap pattern [4,4] is zero. Because 4%6=4%6=4 The total count of prime gap pattern [10,28] is zero. Because 10%6=28%6=4 The total count of prime gap pattern [gapC%6=4,gapD%6=4] is zero. GapRule The total count of prime gap pattern [2,4,2,4,2] is one. show TFTFT Prime5 cut all, except initial. [2,4,2,4,2] pay6 is [,T,F,T,F,T] The total count of prime gap pattern [4,2,4,2,4,6] is one.show FTFTF1 Prime7 cut all, except initial. [4,2,4,2,4,6] pay6 is [,F,T,F,T,F,1]

```<a name=index>
aa()
■■　composite number, prime number
■■　primeArr[pID] example
■■　Prime sequence and GAP sequence
■■　GAP sequence in pay6 form explained
■■　gap pattern 2,26 is forbidden forever
■■　gap pattern 2,4,2,4,2 occur only once
■■　gap pattern 4,2,4,2,4,6 occur only once
■■　Composite has NO RIGHT to issue prime gap
■■　【  debug  】 code for general case
■■　pay6 string left end a comma, ',T,1,F,T'
■■　pay6 string right end no comma
■■　[I1,g1,I2,g2,I3] I1 must be a prime
■■　[,T,F,T,F,T] happen once only, conclude
■■　[,F,T,F,T,F,1] happen once , conclude
■■　First reason use pay6 form
■■　Second reason use pay6 form
■■　Six n prime table, sample
■■　Six n prime table to Six n harmony rule
■■　pay6 prime table, sample
■■　pay6 prime table to Six n harmony rule
■■　freeman2.com/pay6get2.htm
■■　pay6get2.htm build table
■■　pay6nxt2.htm has prime data from 2 to 60000011
■■　freeman2.com/pay6nxt2.htm NXT
■■　pay6nxt2.htm motivation
■■　pay6nxt2.htm major work
■■　TF[][], FT[][] group same pattern
■■　continuous p6tf="" RUN j2
■■　T gap change 6*m-1 prime to 6*n+1 prime
■■　Z gap is prime_gap/6 remainder=0
■■　T gap is prime_gap/6 remainder=2
■■　F gap is prime_gap/6 remainder=4
■■　pay6 string 59962211,5,1t,2,1,2f,
■■　pay6nxt2.htm 8,914,300; pay6get2.htm 562,472
■■　build primeArr[] 2,3,5 ... 60000011
■■　Box32 study next gap section
■■　from integer to prime factorization
■■　pay6 string look back pay6 pattern
■■　twin prime ,T, ; cousin prime ,F,
■■　small prime study, scissors5,7,11
■■　Given primeID  find prime.
■■　input integers, click

<a name=docB001>
aa()
2018-11-24-19-53
tute0070.htm and pay6get2.htm and pay6nxt2.htm
do not have new concept, but use an uncommon
prime gap notation called pay6=pace six. Because
this file consider primes in 6*m+1 and 6*n-1
form. Six in pace six come from 6*m+1 and 6*n-1
Primes are odd numbers, prime3 is sharp cutter.
Prime in pay6=pace six form automatically drop
prime2 and prime3 factors.
Six n harmony rule=avoid_Prime3_cut rule
<a name=docB002>
This file
http://freeman2.com/tute0070.htm
is document for
http://freeman2.com/pay6get2.htm and
http://freeman2.com/pay6nxt2.htm
explain why write prime gap=14 as 2T?
explain why write pay6nxt2.htm Box32 output as
<a name=docB003>
[[
i0=pID=24,prime=97 ; [1T,F,T,F,T,F]
89,97,101,103,107,109,113:
do.more.here; j0=29, pay6million[j0]=[F] p_new=113
future candidate prime is one of next (tf0=2)
113=113; gap=0, pay6 gap=0 ===== prime
115=5*23; gap=2, pay6 gap=0T
119=7*17; gap=6, pay6 gap=1
121=11*11; gap=8, pay6 gap=1T
125=5*5*5; gap=12, pay6 gap=2
127=127; gap=14, pay6 gap=2T ***** prime
]]
<a name=docB004>
explain why write pay6nxt2.htm Box33 output as
[[
2604499,1F,2,T,1,4F,T,4,1F,1T,1F,1,5,2T,1F,1,1,3T,4F,T,F,T,F,4T
2988707,2T,1F,5,1,2,10T,1,2F,2T,3F,T,1,F,6T,2,7,1,3F,T,F,T,F,4T
5937553,1,3,1F,1,1,5,4,2T,1F,1,2T,1F,4,1T,3F,5,4T,5F,T,F,T,F,4T
6631591,2F,2,3,1,3,T,2F,2T,F,T,3F,2,1,4,4T,13F,2,3,T,F,T,F,4T
12172469,T,1,F,1,4T,6,5,2,2F,T,6F,2,5,3,1T,1,1,3F,T,F,T,F,4T
15520471,3,1F,2T,F,3T,3F,1T,2,8,1,3,2F,4,T,2F,7T,1,3F,T,F,T,F,4T
16453687,1,8,5,1F,T,2F,2T,3,3,4,2F,8T,2,1F,1,1,4T,3F,T,F,T,F,4T
16727797,3F,1T,1,5,4F,1,3T,2,1F,1T,5F,1,2T,1,F,3,2T,4F,T,F,T,F,4T
23639093,2T,F,2,1T,1,2,1F,12T,1F,2T,3F,T,4F,T,2,F,3T,5F,T,F,T,F,4T
24786121,2,F,2,2,T,1F,3T,1,3,6,5,F,2T,8F,2T,F,4T,F,T,F,T,F,4T
39674449,4,F,1,3,3T,3F,3,2,4,2T,3,9,4,4,1,F,3,2,T,F,T,F,4T
41213771,T,3,3F,3T,1,4,1F,8T,F,T,3,1F,6,2T,2F,1,6T,F,T,F,T,F,4T
47272601,1,4T,4,1,1,2,9F,8,T,F,2T,4,2F,10T,1F,4,2T,5F,T,F,T,F,4T
49610419,3F,12,2T,1F,1,1T,1,3F,12,3,2T,9F,5T,1F,3,6,2,12,T,F,T,F,4T
55057559,1T,7,3,5,F,T,6F,6T,6,1F,T,1F,3,T,F,6T,6,F,T,F,T,F,4T
57812143,5F,3T,2F,5,3T,1F,6T,2F,T,F,9,4,T,2F,2,1T,3,F,T,F,T,F,4T
]]
Reader should verify these output are correct?
before study the result.
<a name=docB005>
Integer 2200 can be prime factorized as
2*2*2*5*5*11. This number 2200 is composite
number.
If an integer can be prime factorized
only as one multiply itself, this number
is a prime number. First ten primes are
2,3,5,7,11,13,17,19,23,29
http://freeman2.com/pay6nxt2.htm stored
3562116 primes from 2,3 to 60000011 .
<a name=docB006>
aa()
Prime gap gk is defined as
gk = pk+1 - pk ---eq.P01
re-write as
pk + gk = pk+1 ---eq.P02
<a name=docB007>
Primes  2, 3, 5, 7, 11,13,17,19,23,29 .....
has gap 1, 2, 2, 4,  2, 4, 2, 4, 6, .....
Prime and gap sum are
2 + 1 = 3
3 + 2 = 5
5 + 2 = 7
7 + 4 =11
11 + 2 =13
13 + 4 =17
17 + 2 =19
19 + 4 =23
23 + 6 =29
.....

<a name=docB008>
freeman2.com all prime files use primeArr[]
as prime array.
primeArr[0]=prime2, primeID=0 for prime2
primeArr[1]=prime3, primeID=1 for prime3
primeArr[2]=prime5, primeID=2 for prime5
primeArr[3]=prime7, primeID=3 for prime7
primeArr[4]=prime11,primeID=4 for prime11
primeArr[5]=prime13,primeID=5 for prime13
.....
primeArr[20]=prime73,pID=20 for prime73
.....
primeArr[3562115]=prime60000011, that is
pID=3562115 for prime60000011 . Larger
primes are not stored in pay6nxt2.htm.

<a name=docB009>
primeID short as pID. primeArr[pID]=prime
pID can be even, can be 5,15,25 etc. but
primeArr[pID] must not have factor 2,3,5,7
etc. So a set of number is prime? or is pID?
it is easy to distinguish.
2018-11-24-21-00

<a name=docB010>
2018-11-30-09-31
Prime sequence is
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53 ...
Prime GAP sequence is
1,2,2,4,2,4,2,4,6,2,6,4,2,4,6, ...
GAP sequence in pay6 form is
2,o,T,T,F,T,F,T,F,1,T,1,F,T,F,1,1,T,1,F, ...
First 2 is prime2, the following are all gaps
'o'=one=gap1=3-2. only odd number gap
'T'=Two=gap2=5-3=19-17=139-137 etc.
'F'=Four=gap4=17-13=47-43=281-277 etc.
'1F'=1*6+Four=gap10=149-139=7187-7177 etc.
'2T'=2*6+Two =gap14=127-113=7207-7193 etc.
'3T', '4F' etc are defined similarly.

<a name=docB011>
aa()
At the initial stage 2,3,5,7,11, there are
several unique gap patterns which happen once
then NEVER occur again. See the following
Prime 2 has gap 1=3-2. Odd number gap 1 occur
just once, all future gaps are even.
Prime 3 has gap pattern 2,2 (5-3=2, 7-5=2).
Which occur just once. gap pattern 2,2 is
forbidden forever.
gap pattern 4,4 is forbidden forever.
gap pattern 2,26 is forbidden forever.
gap pattern 4,16 is forbidden forever.
<a name=docB012>
Why forbidden forever? See gaps 2,26
Because gap2,gap26 neighbor three integers,
one of them must contain factor 3.
2969,2971,2997 has gap2,gap26. 2969+2=2971
and 2971+26=2997=3*3*3*3*37
primes 2969,2971, has gap2,gap28 (not gap26).
2971+28=2999=prime. 2%6=2=T and 28%6=4=F
Four after Two and Two after Four; Two and
Four show up in turn that is harmony. See
Six n harmony rule  = avoid_Prime3_cut rule.

<a name=docB013>
Prime 5 has gap pattern [2,4,2,4,2]. In pay6
form, the pattern is [,T,F,T,F,T] This
pattern [,T,F,T,F,T] occur just once. gap
pattern [,T,F,T,F,T] is forbidden forever.
See why goto <a name=docB058> and conclusion
http://freeman2.com/pay6nxt2.htm#bx81
click button . prime 5 javascript code
show up in in Box81. This code ONLY use prime5
to check. Because 5 void [,T,F,T,F,T] forever.
click
output to Box82. Output string contain
<a name=docB014>
"Suggest number near 5" which mean gap
pattern [2,4,2,4,2] or pattern [,T,F,T,F,T]
start from prime5 primes [5,7,11,13,17,19]
has gap [2,4,2,4,2] and never again. Box82
output never see "Suggest number near who"
again.
Program change gap [2,4,2,4,2] to
sum array s=[0,2,6,8,12,14]
s[0]= 0:in [2,4,2,4,2] no sum, initial 0
s[1]= 2:in [2,4,2,4,2] copy left most 2
s[2]= 6:in [2,4,2,4,2] sum 2+4=6
s[3]= 8:in [2,4,2,4,2] sum 2+4+2=8
s[4]=12:in [2,4,2,4,2] sum 2+4+2+4=12
s[5]=14=in [2,4,2,4,2] sum 2+4+2+4+2=14.

<a name=docB015>
After build s=[0,2,6,8,12,14]
sum  0 : prime5 add  0 get prime5
sum  2 : prime5 add  2 get prime7
sum  6 : prime5 add  6 get prime11
sum  8 : prime5 add  8 get prime13
sum 12 : prime5 add 12 get prime17
sum 14 : prime5 add 14 get prime19
sum array s=[0,2,6,8,12,14] simplify calculation.

<a name=docB016>
aa()
To compare the result, in Box81 change code
line from
s=[0,2,6,8,12,14] ;
to
s=[0,2,6,8,12] ;
not care [2,4,2,4,2] sum all over,
effectively care only [2,4,2,4] drop last 2.
relax pattern limitation from [2,4,2,4,2]
to [2,4,2,4] , click  Box82
output has many "Suggest number near who"

<a name=docB017>
Above prime5 void [,T,F,T,F,T] forever.
Below prime7 void [,F,T,F,T,F,1] forever.
Start from prime 7, it has gap pattern
[4,2,4,2,4,6]. In pay6 form, the pattern
is [,F,T,F,T,F,1] How to get [4,2,4,2,4,6]?
4=prime11-prime7,  pay6 F=Four=4 gap4
2=prime13-prime11, pay6 T=Two=2  gap2
4=prime17-prime13, pay6 F=Four=4 gap4
2=prime19-prime17, pay6 T=Two=2  gap2
4=prime23-prime19, pay6 F=Four=4 gap4
6=prime29-prime23. pay6 1=1*6=6  gap6
In [,F,T,F,T,F,1] pay6 1=gap6

<a name=docB018>
This pattern  [,F,T,F,T,F,1] occur just
once. gap pattern [,F,T,F,T,F,1] never
show up again. See why goto <a name=docB069>
and conclusion
http://freeman2.com/pay6nxt2.htm#bx81
click button . prime 7 javascript code
show up in in Box81. click
output to Box82. Output string contain
<a name=docB019>
"Suggest number near 7" which mean gap
pattern [4,2,4,2,4,6] or pay6 pattern
[,F,T,F,T,F,1] start from prime7 primes
[7,11,13,17,19,23,29] has gap [4,2,4,2,4,6]
and never again.
Box82 output other part never see
"Suggest number near who" again.
<a name=docB020>
To compare the result, in Box81 change
code line from
s=[0,4,6,10,12,16,22];
to
s=[0,4,6,10,12,16];
relax pattern limitation, click
Box82 output has many "Suggest number near who"
"Suggest" indicate this suggestion may or
may NOT be a prime/gap pattern. Code exam
a number use only number divide by 7 and
other primes are not checked.
2018-11-30-10-25

<a name=docB021>
aa()
To see extended range,
not use   s=[0,4,6,10,12,16,22];
use relax s=[0,4,6,10,12,16]; for more match
and modify from
nbgn=1
nend=100
to
nbgn=10000
nend=10100
n in nbgn and nend is used in 6*n+1
and 6*n-1. if n=10000, integer=60001.

<a name=docB022>
output has
[[
Suggest number near 60031=173*347
Suggest number near 60073=13*4621
Suggest number near 60115=5*11*1093
Suggest number near 60157=43*1399
Suggest number near 60199=37*1627
Suggest number near 60241=107*563
Suggest number near 60283=23*2621
Suggest number near 60325=5*5*19*127
Suggest number near 60367=17*53*67
Suggest number near 60409=193*313
Suggest number near 60451=61*991
Suggest number near 60493=60493
Suggest number near 60535=5*12107
Suggest number near 60577=11*5507
]]

<a name=docB023>
Suggest number near 60577 but
60577=11*5507
60577 is not a prime, how can it produce
first gap 4 in [4,2,4,2,4,6] ?
Only prime number can say its gap is 4.
Composite number has NO RIGHT to do so.
Except one, other are all FAKE answer.
only
Suggest number near 60493=60493  start from
a prime60493 To check a given gap sequence
[4,2,4,2,4,6] must make sure all integer in
this sequence are primes. Whether start prime
60493 follow gap pattern is [4,2,4,2,4,6] ?
exam next.

<a name=docB024>
Suggest number near 60493
60493 is a prime number
60493 section is next
[[
n=10082,(6*n+1+0)%7=6
n=10082,(6*n+1+4)%7=3
n=10082,(6*n+1+6)%7=5
n=10082,(6*n+1+10)%7=2
n=10082,(6*n+1+12)%7=4
n=10082,(6*n+1+16)%7=1
Suggest number near 60493
]]
<a name=docB025>
Above lines has "%7", mean ONLY checked
prime7 factor. Other prime factor not checked.
Paste "60493,F,T,F,T,F,1" to pay6nxt2.htm
Box22 and click  get
60493,60497,60499,60503,60505,60509,60515
"60515" is not a prime. Therefore
Suggest number near 60493=60493
This suggestion is false. See <a name=docB069>
for gap sequence ",F,T,F,T,F,1".

<a name=docB026>
aa()
60493 section has gap match numbers as next
[[
n=10082,(6*n+1+0)=60493
n=10082,(6*n+1+4)=60497
n=10082,(6*n+1+6)=60499
n=10082,(6*n+1+10)=60503
n=10082,(6*n+1+12)=60505
n=10082,(6*n+1+16)=60509
]]
<a name=docB027>
They must ALL be primes, then the
prime gap sum
given s=[0,4,6,10,12,16,22];
relax s=[0,4,6,10,12,16];
is qualified as prime gap sum;

<a name=docB028>
write simple code
[[
n=10082
6*n+1+0
6*n+1+4
6*n+1+6
6*n+1+10
6*n+1+12
6*n+1+16
]]

<a name=docB029>
Real number calculator output next
[[
6*n+1+0
60493
6*n+1+4
60497
6*n+1+6
60499
6*n+1+10
60503
6*n+1+12
60505
6*n+1+16
60509
]]

<a name=docB030>
but
6*n+1+0
60493     =60493*1 prime
6*n+1+4
60497     =60497*1 prime
6*n+1+6
60499     =101*599 composite
6*n+1+10
60503     =17*3559 composite
6*n+1+12
60505     =5*12101 composite
6*n+1+16
60509     =60509*1 prime

<a name=docB031>
aa()
Not ALL be prime, then
Suggest number near 60493 (60493 is prime)

<a name=docB032>
prime7 cut pay6 string [,F,T,F,T,F,1] .
In [,F,T,F,T,F,1] drop right most 1(=1*6)
prime7 not cut pay6 string [,F,T,F,T,F]
and [,F,T,F,T,F] show up many times in
small prime section.
If tighten [,F,T,F,T,F] by adding one ',1'
at right end, result [,F,T,F,T,F,1]
prime7 does cut pay6 string [,F,T,F,T,F,1]
that is prime7 cut gap [,4,2,4,2,4,6]
prime7 cut sum sequence [0,4,6,10,12,16,22]
[,F,T,F,T,F,1] show up once, then never
appear.

<a name=docB033>
In [,F,T,F,T,F,1] drop last 1*6, that is
in [0,4,6,10,12,16,22] drop last sum 22 get
Relaxed pay6 [,F,T,F,T,F]
Relaxed gap  [,4,2,4,2,4]
Relaxed sum  [0,4,6,10,12,16]
do show up many times. Why pay6 [,F,T,F,T,F]
not show up for n=10000 (in 6*n+1)? Because
nbgn=10000
nend=10100
not cover one relaxed gap.
<a name=docB034>
Prime 60493 is greater than prime7,23,61
Relaxed SMALL gap  [,4,2,4,2,4] show up
in SMALL prime section and in greater prime
section (around 60493), long string small
gap [,4,2,4,2,4] is unlikely.
Liu,Hsinhan self explain, no proof.
n=9979566 has one gap [,4,2,4,2,4] //a712091548
59877397,59877401,59877403,59877407,59877409,59877413
2018-11-30-11-25

<a name=docB035>
2018-11-30-15-49
why pay6 string left end a comma,
but right end no comma ?
In
http://freeman2.com/pay6nxt2.htm#bx81
```
Click 【 debug
```
<a name=docB036>
aa()
Box81 has four lines
[[
//below pay6 is prime gap pay6 string
//string left end a comma, right end no comma.
pay6=',1T,2F,3T,4F,T,F';
//above user control parameter. //below program code.
]]
why string left end a comma, right end no comma ?
<a name=docB037>
consider primes
23,29,31,37,41,43,47
it has gaps
6, 2, 6, 4, 2, 4
it has pay6
1, T, 1, F, T, F
pay6=',1,T,1,F,T,F';

define gapSum as [0,6,8,14,18,20,24]
where 8=6+2; 14=6+2+6; 18=6+2+6+4;
20=6+2+6+4+2; 24=6+2+6+4+2+4;

<a name=docB038>
Prime 23=(6*n-1) when n=4
with (6*n-1) in hand and calculation come to n=4
use a for(i0) loop go over gapSum[i0] to create
integers 23,29,31,37,41,43,47
n=4;
for(i0=0;i0<7;i0++)number[i0]=(6*n-1)+gapSum[i0];
After for(i0) loop, number[i0] array store
[23,29,31,37,41,43,47]

<a name=docB039>
recall gapSum=[0,6,8,14,18,20,24]
In this creation process, need gapSum[0]=0
such that number[0]=(6*4-1)+gapSum[0]=23;
If define pay6=',1,T,1,F,T,F';
command p6=pay6.split(',') change string pay6
to array p6=['','1','T','1','F','T','F']
Code change left most empty string '' to zero.
p6[] is then convert to gap g=[0,6,2,6,4,2,4]
g[] is summed to s=[0,6,8,14,18,20,24]
s[] is actually used in calculation. See code
ans=(6*n+sixnPM1+s[iq])/p[ip];

<a name=docB040>
Above explain pay6=',1,T,1,F,T,F'; left end
need one comma to provide empty seat for g[0]=0;
Next explain why pay6=',1,T,1,F,T,F'; right end
cannot have a comma. If write input line as
pay6=',1,T,1,F,T,F,'; //right end comma wrong
with a right end comma, command
p6=pay6.split(',') change string pay6
to array p6=['','1','T','1','F','T','F','']
<a name=docB041>
aa()
p6.length change from 7 to 8. for(i0) loop
run one more loop, but s=[0,6,8,14,18,20,24]
no eighth element, for(i0) loop calculation
cause error.
Right end comma or left end comma influence
p6=pay6.split(',') result. If it is not an
input pay6 string, a searching pay6 string
contain both end comma make search clear.
For example search ",T,F,1," refuse match
with ",T,F,1T".
2018-11-30-16-38

<a name=docB042>
2018-11-30-18-52
Prime number has two different flavors.
One is 6*n+1 type primes, 7,13,19,31,37 etc.
Two is 6*n-1 type primes, 5,11,17,29,41 etc.
If given pay6A=',1T,2F,3T,4F,T,F' what is its
first prime? pay6A first element is 1T for
1*6+Two=8.
A T gap (1T,2T,3T etc are the same) change
from 6*n-1 type prime to 6*n+1 type prime.
<a name=docB043>
For example prime gap 2 between 11 and 13,
gap T=2 change 6*2-1 to 6*2+1.
prime gap 2T=2*6+Two=14 between 113 and 127
gap 2T=2*6+Two=14 change 6*19-1 to 6*21+1.
With this understanding, the given pay6 string
pay6A=',1T,2F,3T,4F,T,F' must start from
6*n-1 type prime, if did it wrong, choose
6*n+1 type prime, the following integers have
factor 3 involve and following integers are
alien.

<a name=docB044>
Another example
pay6B=',1F,2T,3F,T,;318979,318986';
n=318979, to n=318986
n=318982 is first match
Since pay6B first element 1F is a F type gap,
A F gap (F,1F,2F,3F etc are the same) change
from 6*m+1 type prime to 6*n-1 type prime.
For example prime gap 1F=10 between 139 and 149,
gap 1F=10 change 139=6*23+1 to 6*25-1=149.
given pay6B=',1F,2T,3F,T' must start from
6*n+1 type prime to avoid prime 3 cut.

<a name=docB045>
2018-11-30-19-14
pay6nxt2.htm Box34
n=318982 [1F,2T,3F,T] exist. Suggest number near 1913893
click button  give the following input data.
[[
,1F,2T,3F,T;318979,318986
]]
<a name=docB046>
aa()
';' separate ',1F,2T,3F,T;318979,318986' to two
parts. Left side is ',1F,2T,3F,T', right side
is '318979,318986'.
',1F,2T,3F,T' is pay6 string under study.
'318979,318986' is n value in 6*n+1 or 6*n-1.
nBegin=318979, nEnd=318986.
<a name=docB047>
Because ',1F,2T,3F,T' left end has F gap.
F gap change 6*m+1 prime to 6*n-1 prime. Code
choose 6*m+1 and substitute m=318979, m=318980
until m=318986 create 6*m+1 integers. If reader
see m=100000, expect 6*m+1 integer be 600001.
Integer is six time greater than n value.
Liu,Hsinhan confused once.
<a name=docB048>
First iteration m=318979 and 6*m+1=1913875
Start from 1913875, use gap ',1F,2T,3F,T' to
build the following integers.
1913875+1F=1913875+1*6+Four=1913885
1913875+1F+2T=1913885+2*6+Two=1913899
1913875+1F+2T+3F=1913899+3*6+Four=1913921
1913875+1F+2T+3F+T=1913921+Two=1913923
<a name=docB049>
Code divide
1913875,1913885,1913899,1913921,1913923
with user supplied primes 5,7,11,13,17,19
If one whole division (remainder=0) then
test fail. If all division have decimal
fraction, then
1913875,1913885,1913899,1913921,1913923
not contain primes 5,7,11,13,17,19
Second iteration change m=318979 to m=318980.
Repeat same procedure.

<a name=docB050>
//n=318982 first match
When m=318982, 6*m+1=1913893
1913893+1F=1913893+1*6+Four=1913903
1913893+1F+2T=1913903+2*6+Two=1913917
1913893+1F+2T+3F=1913917+3*6+Four=1913939
1913893+1F+2T+3F+T=1913939+Two=1913941
get five integers, copy paste them
1913893,1913903,1913917,1913939,1913941
to http://freeman2.com/pay6nxt2.htm#bx23
<a name=docB051>
aa()
Box23 and click  output to
Box23. Prime number do not change.
Composite number become 1913943=3*653*977
If output do not have '=' then all integers
are primes. Conclude when m=318982,
6*m+1=1913893 match ',1F,2T,3F,T' is true.

<a name=docB052>
Earlier document say checked small primes
5,7,11,13,17,19 Suggest match may contain
factor of 23,29 Now it is possible check
all prime factorization for prime<60000011.
2018-12-06-17-40 finished click button
. //a712061928

<a name=docB053>
Click  Box34 output has
[[
n=318982 [1F,2T,3F,T] exist. Suggest number near 1913893
But number may contain factor 23,29 etc.
]]
<a name=docB054>
both are suggestion, and both are fake answer.
Because in
primes 23,29,31,37,41,43,47
gaps     6, 2, 6, 4, 2, 4
pay6     1, T, 1, F, T, F
pay6='1, T, 1, F, T, F' suggested every integer
23,29,31,37,41,43,47 must ALL be primes. If any
one is not prime, this n and this pay6 string
failed to find a match.
<a name=docB055><a name=errBgn>error begin
Most important is the first number
(23 in 23,29,31,37,41,43,47) must be a prime.
If not, then it has no right to issue first gap.
For example given
gaps     4, 2, 6, 4, 2, 4
pay6     F, T, 1, F, T, F
<a name=docB056>
aa()
when n loop to n=4 get 6*n+1=6*4+1=25 (5*5)
25+4=29 a prime
29+2=31 a prime
31+6=37 a prime
37+4=41 a prime
41+2=43 a prime
43+4=47 a prime
All following integers are prime, but
the start number 25 is not a prime,
25 has no right to issue gap 4=F .

<a name=docB057>
Start number must be a prime, this fact is
used to show that prime5 cut
pay6=',T,F,T,F,T' except first occurence.
Start number must be a prime, this fact is
used to show that prime7 cut
pay6=',F,T,F,T,F,1' except first occurence.

<a name=docB058>
2018-11-30-19-43
Next is three days ago work
2018-11-27-20-57
from 2 to 60000011 [,T,F,T,F,T]
happen once, then never, why never?
first pay6 is T,
T change from 6*n-1 to 6*n+1
consider start from 6*n-1
6*n-1+0         //sum 0
6*n-1+2         //sum 2
6*n-1+2+4       //sum 6
6*n-1+2+4+2     //sum 8
6*n-1+2+4+2+4   //sum12
6*n-1+2+4+2+4+2 //sum14
sum=[0,2,6,8,12,14]
<a name=docB059>
consider
6*n-1
6*n+1
6*n+5
6*n+7
6*n+11
6*n+13
prime 5 cut who:
25%5=0
27%5=2
29%5=4
31%5=1
33%5=3
35%5=0
then
<a name=docB060>
(6*n-1)%5 has five possibility
(6*n-1)%5=0
(6*n-1)%5=1
(6*n-1)%5=2
(6*n-1)%5=3
(6*n-1)%5=4

<a name=docB061>
aa()
if (6*n-1+0)%5=0 then ZERO
(6*n-1+2)%5=2 first remainder 2
(6*n-1+6)%5=1
(6*n-1+8)%5=3
(6*n-1+12)%5=2 second remainder 2
(6*n-1+14)%5=4
<a name=docB062>
example
1865,T,F,T,F,T
1865,1867,1871,1873,1877,1879
if (6*n-1)%5=0 , ( 1865%5=0 ) then
prime5 not cut [,T,F,T,F,T]
this error line was written
on 2018-11-27-21-??
see Start number must be a prime
see errBgn (error begin).

<a name=docB063>
2018-11-27-21-12
if (6*n-1)%5=1 then
(6*n-1+2)%5=3 first remainder 3
(6*n-1+6)%5=2
(6*n-1+8)%5=4
(6*n-1+12)%5=3 second remainder 3
(6*n-1+14)%5=0 ZERO prime5 cut [,T,F,T,F,T]
if (6*n-1)%5=1, prime5 cut (6*n-1+14)
example
[,T,F,T,F,T]
41,43,47,49,53,55
prime5 cut [,T,F,T,F,T] at 55

<a name=docB064>
2018-11-27-21-27
if (6*n-1)%5=2 then
(6*n-1+2)%5=4 first remainder 4
(6*n-1+6)%5=3
(6*n-1+8)%5=0 ZERO prime5 cut [,T,F,T,F,T]
(6*n-1+12)%5=4 second remainder 4
(6*n-1+14)%5=1
if (6*n-1)%5=2, prime5 cut (6*n-1+8)
example
[,T,F,T,F,T]
77,79,83,85,89,91
prime5 cut [,T,F,T,F,T] at 85

<a name=docB065>
2018-11-27-21-30
if (6*n-1)%5=3 then
(6*n-1+2)%5=0 ZERO prime5 cut [,T,F,T,F,T]
(6*n-1+6)%5=4
(6*n-1+8)%5=1
(6*n-1+12)%5=0 ZERO prime5 cut [,T,F,T,F,T]
(6*n-1+14)%5=2
if (6*n-1)%5=3, prime5 cut (6*n-1+2)
example
[,T,F,T,F,T]
83,85,89,91,95,97
prime5 cut [,T,F,T,F,T] at 85,95

<a name=docB066>
aa()
2018-11-27-21-33
if (6*n-1)%5=4 then
(6*n-1+2)%5=1 first remainder 1
(6*n-1+6)%5=0 ZERO prime5 cut [,T,F,T,F,T]
(6*n-1+8)%5=2
(6*n-1+12)%5=1 second remainder 1
(6*n-1+14)%5=3
if (6*n-1)%5=4, prime5 cut (6*n-1+6)
<a name=docB067>
example
[,T,F,T,F,T]
59,61,65,67,81,83
prime5 cut [,T,F,T,F,T] at 65

<a name=docB068>
2018-11-27-21-40
if (6*n-1)%5=0, prime5 not cut [,T,F,T,F,T]
see Start number must be a prime
see errBgn (error begin).

if (6*n-1)%5=0, prime5 cut at (6*n-1+0)
if (6*n-1)%5=1, prime5 cut at (6*n-1+14)
if (6*n-1)%5=2, prime5 cut at (6*n-1+8)
if (6*n-1)%5=3, prime5 cut (6*n-1+2) +12 too
if (6*n-1)%5=4, prime5 cut at (6*n-1+6)
put sum array to compare sum=[0,2,6,8,12,14]
2018-11-30-19-58 done include prime5

<a name=TFTFT>

The total count of prime gap pattern
[2,4,2,4,2] is one. pay6 is [,T,F,T,F,T]

prime gap pattern [,T,F,T,F,T] every
possibility cut by prime5.
Prime5 cut 25, cut 55, cut 95, etc. but
prime5 not cut integer 5. This pattern
[,T,F,T,F,T] show up once at prime5
5,7,11,13,17,19 and never show up again.

<a name=docB069>
2018-11-30-20-00
Next is two days ago work

prime7 has gap ,F,T,F,T,F,1, never occur later
prime7 has gap ,T,F,T,F,T,F, never occur later
",T,F,T,F,T,F," cut by prime5, because
",T,F,T,F,T" cut by prime5.
2018-11-28-17-14
choose
prime7 has gap ,F,T,F,T,F,1, never occur later

<a name=docB070>
2018-11-28-17-21
from 2 to 60000011 [,F,T,F,T,F,1]
happen once, then never, why never?
first pay6 is F=Four=4,
F change from 6*n+1 to 6*m-1=6*(n+1)-1

consider start from 6*n+1
6*n+1+0            //sum 0
6*n+1+4            //sum 4
6*n+1+4+2          //sum 6
6*n+1+4+2+4        //sum10
6*n+1+4+2+4+2      //sum12
6*n+1+4+2+4+2+4    //sum16
6*n+1+4+2+4+2+4+6  //sum22
sum=[0,4,6,10,12,16,22]
<a name=docB071>
aa()
consider
6*n+1+0
6*n+1+4
6*n+1+6
6*n+1+10
6*n+1+12
6*n+1+16
6*n+1+22

<a name=docB072>
prime7 cut who?
21%7=0
23%7=2
25%7=4
27%7=6
29%7=1
31%7=3
33%7=5
35%7=0
then
<a name=docB073>
(6*n+1)%7 has seven possibility
(6*n+1)%7=0
(6*n+1)%7=1
(6*n+1)%7=2
(6*n+1)%7=3
(6*n+1)%7=4
(6*n+1)%7=5
(6*n+1)%7=6

<a name=docB074>
2018-11-28-17-31
if (6*n+1+0)%7=0 then ZERO
(6*n+1+4)%7=4
(6*n+1+6)%7=6
(6*n+1+10)%7=3
(6*n+1+12)%7=5
(6*n+1+16)%7=2
(6*n+1+22)%7=1
if (6*n+1)%7=0 prime7 cut [,F,T,F,T,F,1]
at (6*n-1+0)
example n=15,(6*n+1+0)%7=0 ###cut by 7
[,F,T,F,T,F,1]
91,95,97,101,103,107,113  91=7*13
prime7 cut [,f,T,F,T,F,1] before enter f

<a name=docB075>
//2018-11-28-17-42
var i0,j0,k0,L0,m0,n0;
nbgn=10
nend=1000
g=[0,4,6,10,12,16,22]; //a711281752
ou=''; //a711281742
for(n=nbgn;n<nend;n++){cut=0;
for(gID=0;gID<g.length;gID++){
j0=((6*n+1+g[gID])%7);if(!j0)cut=1;
ou+='n='+n+',(6*n+1+'+g[gID]+')%7='+j0
+(j0==0?' ###cut by 7':'')+'\n'
} //for(gID=0;gID<g.length;gID++)
//ou+=(cut?'\n':'\$\$ Suggest number near '     +(6*n+1)+'\n\n')
ou+=(cut?'\n':'Suggest number near '+(6*n+1)+'\n\n')
ou+='\n'
} //for(n=nbgn;n<nend;n++) //a711281758
ou

<a name=docB076>
aa()
2018-11-28-19-21
if (6*n+1+0)%7=1 then
(6*n+1+4)%7=5
(6*n+1+6)%7=0 ZERO
(6*n+1+10)%7=4
(6*n+1+12)%7=6
(6*n+1+16)%7=3
(6*n+1+22)%7=2
if (6*n+1)%7=1 prime7 cut [,F,T,F,T,F,1]
at (6*n+1+6)
example n=21,(6*n+1+6)%7=0 ###cut by 7
[,F,T,F,T,F,1]
127,131,133,137,139,143,149  133=7*19
prime7 cut [,F,T,f,T,F,1] before enter f

<a name=docB077>
2018-11-28-19-28
if (6*n+1+0)%7=2 then
(6*n+1+4)%7=6
(6*n+1+6)%7=1
(6*n+1+10)%7=5
(6*n+1+12)%7=0 ZERO
(6*n+1+16)%7=4
(6*n+1+22)%7=3
if (6*n+1)%7=2 prime7
cut [,F,T,F,T,F,1] at (6*n+1+12)
<a name=docB078>
example n=20,(6*n+1+12)%7=0 ###cut by 7
[,F,T,F,T,F,1]
121,125,127,131,133,137,143  133=7*19
prime7 cut [,F,T,F,T,f,1] before enter f

<a name=docB079>
2018-11-28-19-33
if (6*n+1+0)%7=3 then
(6*n+1+4)%7=0 ZERO
(6*n+1+6)%7=2
(6*n+1+10)%7=6
(6*n+1+12)%7=1
(6*n+1+16)%7=5
(6*n+1+22)%7=4
if (6*n+1)%7=3 prime7
cut [,F,T,F,T,F,1] at (6*n+1+4)
<a name=docB080>
example n=19,(6*n+1+4)%7=0 ###cut by 7
[,F,T,F,T,F,1]
115,119,121,125,127,131,137,  119=7*17
prime7 cut [,F,t,F,T,F,1] before enter t

<a name=docB081>
aa()
2018-11-28-19-40
if (6*n+1+0)%7=4 then
(6*n+1+4)%7=1
(6*n+1+6)%7=3
(6*n+1+10)%7=0 ZERO
(6*n+1+12)%7=2
(6*n+1+16)%7=6
(6*n+1+22)%7=5
if (6*n+1)%7=4 prime7
cut [,F,T,F,T,F,1] at (6*n+1+10)
<a name=docB082>
example n=18,(6*n+1+10)%7=0 ###cut by 7
[,F,T,F,T,F,1]
109,113,115,119,121,125,131,  119=7*17
prime7 cut [,F,T,F,t,F,1] before enter t

<a name=docB083>
2018-11-28-19-44
if (6*n+1+0)%7=5 then
(6*n+1+4)%7=2
(6*n+1+6)%7=4
(6*n+1+10)%7=1
(6*n+1+12)%7=3
(6*n+1+16)%7=0 ZERO
(6*n+1+22)%7=6
if (6*n+1)%7=5 prime7
cut [,F,T,F,T,F,1] at (6*n+1+16)
<a name=docB084>
example n=17,(6*n+1+16)%7=0 ###cut by 7
[,F,T,F,T,F,1]
103,107,109,113,115,119,125,  119=7*17
prime7 cut [,F,T,F,T,f,1] before enter f

<a name=docB085>
2018-11-28-19-49
if (6*n+1+0)%7=6 then
(6*n+1+4)%7=3
(6*n+1+6)%7=5
(6*n+1+10)%7=2
(6*n+1+12)%7=4
(6*n+1+16)%7=1
(6*n+1+22)%7=0 ZERO
if (6*n+1)%7=6 prime7
cut [,F,T,F,T,F,1] at (6*n+1+22)
<a name=docB086>
aa()
example n=16,(6*n+1+22)%7=0 ###cut by 7
[,F,T,F,T,F,1]
97,101,103,107,109,113,119,  119=7*17
prime7 cut [,F,T,F,T,F,1] before enter 1
2018-11-28-19-53

2018-11-30-20-08 done include prime7

<a name=docB087>
2018-12-01-08-55
for prime gap pattern [,F,T,F,T,F,1]
if (6*n-1)%7=0, prime7 cut at (6*n-1+0)
if (6*n-1)%7=1, prime7 cut at (6*n-1+6)
if (6*n-1)%7=2, prime7 cut at (6*n-1+12)
if (6*n-1)%7=3, prime7 cut at (6*n-1+4)
if (6*n-1)%7=4, prime7 cut at (6*n-1+10)
if (6*n-1)%7=5, prime7 cut at (6*n-1+16)
if (6*n-1)%7=6, prime7 cut at (6*n-1+22)
compare with sum=[0,4,6,10,12,16,22]
<a name=FTFTF1>

The total count of prime gap pattern
[4,2,4,2,4,6] is one.

prime gap pattern [,F,T,F,T,F,1] every
possibility cut by prime7.
Prime7 cut 21, cut 49, cut 77, etc. but
prime7 not cut integer 7. This pattern
[,F,T,F,T,F,1] show up once at prime7
[7,11,13,17,19,23,29] gap [4,2,4,2,4,6]
and never show up again.
2018-12-01-09-11

<a name=docB088>
2018-12-01-14-10
Around 2018-10-2? Liu,Hsinhan think
primes
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47, ...
gaps
1,2,2,4,2,4,2,4,6,2,6,4,2,4,6, ...
write gap in pay6 form
2,o,T,T,F,T,F,T,F,1,T,1,F,T,F,1,1,T,1,F, ...
this pay6 form has great advantage, because

<a name=docB089>
First reason use pay6 form
gap in pay6 form is shorter than in digit
form. For example primes
59999809,59999833,59999837,59999839,59999843,
59999867,59999879,59999881,59999887,59999917,
59999957,59999971,59999981,59999983,59999993,
59999999,60000011
has gaps in digit
59999809,24,4,2,4,24,12,2,6,30,40,14,10,2,10,6,12
change to pay6 form
59999809,4,F,T,F,4,2,T,1,5,6F,2T,1F,T,1F,1,2
<a name=docB090>
pay6 form gap is always shorter than digit gaps
gap  2 change to pay6 T  (Two=2)
gap  4 change to pay6 F  (Four=4)
gap  6 change to pay6 1  (1*6=6)
gap 12 change to pay6 2  (2*6=12)
gap 14 change to pay6 2T (2*6+Two=14)
gap 24 change to pay6 4  (4*6=24)
gap 30 change to pay6 5  (5*6=30)
gap 40 change to pay6 6F (6*6+Four=40)

<a name=docB091>
aa()
Second reason use pay6 form
Six n harmony rule  = avoid_Prime3_cut rule.
The T,F,T,F pattern let Six n harmony rule
paramount display凸顯 T,F,T,F,T,F to forever
To observe Six n harmony rule,
first see http://freeman2.com/prim6n01.htm
second see http://freeman2.com/pay6lst0.htm

<a name=docB092>
prim6n01.htm short section is next ```

 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 5^3 21 127 131 22 7^1 * 19^1 137 23 139 11^1 * 13^1 24 5^1 * 29^1 149 25 151 5^1 * 31^1 26 157 7^1 * 23^1 27 163 167 28 13^2 173 29 5^2 * 7^1 179 30 181 5^1 * 37^1 31 11^1 * 17^1 191 32 193 197 33 199 7^1 * 29^1 34 5^1 * 41^1 11^1 * 19^1 35 211 5^1 * 43^1 36 7^1 * 31^1 13^1 * 17^1 37 223 227 38 229 233 39 5^1 * 47^1 239 40 241
http://freeman2.com/prim6n01.htm
```alert0()
<a name=docB093>
Left column is 6*n-1 prime 11,17,23,29,41 etc.
right column is 6*n+1 prime 13,19,31,37,43 etc.

When prime 11 increase to 13, change from left
column to right column, from 6*2-1 to 6*2+1.
11 increase two become 13.
When prime 13 increase to 17, change from right
column to left column, from 6*2+1 to 6*3-1.
13 increase four become 17.

<a name=docB094>
This two column structure indicate clearly,
left column add 2 move to right column.
right column add 4 move to  left column.
then gap 2,4,2,4 must be in turn. In other
words, gap sequence 2,2 goto no where! and
gap sequence 4,4 goto no where! Six-n prime
table has only two columns, not four column.

gaps 2,2 can never occur (prime
3,5,7 is only initial special case)

gap sequence 4,4 can never occur
and no special case.

<a name=docB095>
After this observation, Liu,Hsinhan realize
Six n harmony rule  = avoid_Prime3_cut rule.
2018-12-01-14-54

2018-12-01-15-58
Above prim6n01.htm six n prime table short section

<a name=docB096>
aa()
Below pay6lst0.htm pay6 prime table short section
```
 p seq.blue:6*n-111,59,83 etc. q seq. gap prime;red:6*n+113,19,43 etc. pIDprime ID pay6string 0 o 1 3 1 o 0 T 2 5 2 T 0 T 2 7 3 T 0 F 4 11 4 F 0 T 2 13 5 T 0 F 4 17 6 F 0 T 2 19 7 T 0 F 4 23 8 F 1 6 29 9 1 0 T 2 31 10 T 1 6 37 11 1 0 F 4 41 12 F 0 T 2 43 13 T 0 F 4 47 14 F 1 6 53 15 1 1 6 59 16 1 0 T 2 61 17 T 1 6 67 18 1 0 F 4 71 19 F 0 T 2 73 20 T 1 6 79 21 1 0 F 4 83 22 F 1 6 89 23 1 1 T 8 97 24 1T 0 F 4 101 25 F 0 T 2 103 26 T 0 F 4 107 27 F 0 T 2 109 28 T 0 F 4 113 29 F 2 T 14 127 30 2T 0 F 4 131 31 F 1 6 137 32 1 0 T 2 139 33 T 1 F 10 149 34 1F 0 T 2 151 35 T 1 6 157 36 1 1 6 163 37 1 0 F 4 167 38 F 1 6 173 39 1 1 6 179 40 1 0 T 2 181 41 T 1 F 10 191 42 1F 0 T 2 193 43 T 0 F 4 197 44 F

http://freeman2.com/pay6lst0.htm
```alert0()
<a name=docB097>
2018-12-01-15-17
In pay6 prime table pay attention to pink column
marked q seq.
o represent gap=1(One), prime3-prime2
T represent gap=2(Two), prime5-prime3
F represent gap=4(Four),prime11-prime7
Except top two T,T (prime 3,5,7; gap 2,2=T,T)
the following is always T,F,T,F in turn.
We never find T,T; We never find F,F;
This T,F in turn is
Six n harmony rule  = avoid_Prime3_cut rule.
2018-12-01-15-21

<a name=docB098>
2018-12-03-08-37
Pay6 Prime number calculator
http://freeman2.com/pay6get2.htm
"get" in file name pay6get2.htm mean
user given prime list (paste to Box21)
click [prime to pay6]
RUN f Box21 in  Box22 out
Program output pay6 string to Box22.

<a name=docB099>
This pay6 string can it restore original
prime list? To answer this question,
click [pay6 to prime]
RUN g Box22 in  Box23 out
original prime list will show up in Box23.
```
<a name=docB100>
 If box22 has pay6 data, click RUN☞i , and click SHOW☞i to see pay6,gap,prime table. To build pay6lst0.htm, paste prime data to box21. click RUN☞f to get pay6 data in Box22. Next, click RUN☞j , save Box24 value to another html file. Open other html file to view table. Modify text string and change ### to your need.
```<a name=docB101>
aa()
2018-12-03-08-55
Above http://freeman2.com/pay6get2.htm
Below http://freeman2.com/pay6nxt2.htm

another program is pay6nxt2.htm
Pay6 small Prime study .

<a name=docB102>
major difference between pay6nxt2.htm and
pay6get2.htm is that
pay6get2.htm no prime definition.
pay6nxt2.htm has prime list from 2,3,5,7,
... to 59999983,59999993,59999999,60000011;
total 3562116 count of primes. They are in
pay6 form. Text file store 3562116 count of
primes. The size is 34,867,065 bytes
<a name=docB103>
34,867,065 prime.count_3562116.txt
8,636,262 prime.count_3562116_pay6_H.txt
Text file store 3562116 count of primes in
pay6 form. The size is 8,636,262 bytes
Size from 34.8 MBytes reduce to 3.8 MBytes.
pay6 string is beneficial.
```
 "nxt" in file name pay6nxt2.htm mean a wish to find next prime gap by gap pattern study, without using integer's prime factorization. But not achieve this goal now, possibly never reach this goal in future. However, if find more prime gap rules, it is possible to cut the candidate integer to much smaller count. Minimize the calculation of integer's prime factorization. 2018-12-03-09-24
```alert0()
<a name=docB104>
2018-12-03-09-32
around 2018-10-20 to 2018-10-24 Liu,Hsinhan
think a problem.
From prime gap list never appear 2,8 pattern
found
Prime Gap Six n harmony rule
http://freeman2.com/pay6nxt2.htm#PrimeGapRules
If find another pattern never show up,
then, I may find another Prime Gap what
rule. hopefail
<a name=docB105>
Base on this thought, Liu,Hsinhan
write code to find prime gap pay6 sequence
0T0F (2,4), 0T1F (2,10), 0T2F (2,16), ...
2T0F (14,4), 3T4F (20,28), 4T6F (26,40),
...
0F0T (4,2), 1F3T (10,20), 2F5T (16,32), ...
0F2T (4,14), 1F5T (10,32), 3F6T (22,38),
...
<a name=docB106>
aa()
list the summation for every possible pattern
see which pattern has total zero count of
appearance. If this nul result show up, it
will help me to find new prime gap rule,
just like prime gap never appear 2,8 helped
me.
2018-10-24 Liu,Hsinhan stopped Frequently
used Chinese characters work
http://freeman2.com/c/index.htm
Change attention to prime gap pay6 work
http://freeman2.com/pay6nxt2.htm

<a name=docB107>
pay6nxt2.htm major work is
RUN j1 and RUN j2 above Box28
```
Next two lines are FOCUS output This file MAIN CONCERN.
RUN j1 , , , show ,
RUN j2 , , , show ,
```<a name=docB108>
2018-12-03-10-02
If click
Code build TF[][], FT[][] two arrays.
TF[][] record T first, F second
0T0F (2,4), 0T1F (2,10), 0T2F (2,16), ...
2T0F (14,4), 3T4F (20,28), 4T6F (26,40),
...

FT[][] record F entering, T depart
0F0T (4,2), 1F3T (10,20), 2F5T (16,32), ...
0F2T (4,14), 1F5T (10,32), 3F6T (22,38),
3F6T (22,38) say pay6 3F is gap22=3*6+Four ;
pay6 6T is gap38=6*6+Two ;

<a name=docB109>
"1F3T" and "1FT3" are one thing,
In "1F3T" F begin, must stop at T.
In "2T0F" T begin, must stop at F.
then "1F3T" simplify to "1F3"
and "2T0F" simplify to "2T0" .
Code output 1F3 (1FT3 or 1F3T) the next
1FT3=53,193,204,258,279,310,418,
1FT3=53 say primeID=53=prime251 has 1F=10
enter 251, 3T=20 exit 251 to following T.
see 241,251,257,263,269,271;
241+1*6+Four=251; 251+6+6+6+Two=271;
http://freeman2.com/pay6nxt2.htm#sumRight
Click button  has more.

<a name=docB110>
Prime251 neighbor primes are next
[[
241 gap 10=1*6+Four=pay6 style 1F
251 gap  6=1*6=pay6 style 1
257 gap  6=1*6=pay6 style 1
263 gap  6=1*6=pay6 style 1
269 gap  2=0*6+Two=pay6 style T
271
]]
251 is target prime.
<a name=docB111>
aa()
RED
241 gap 10=1*6+Four=pay6 style 1F
enter 251
exit from 251 use BLUE
251 gap  6=1*6=pay6 style 1
257 gap  6=1*6=pay6 style 1
263 gap  6=1*6=pay6 style 1
269 gap  2=0*6+Two=pay6 style T
271
<a name=docB112>
exit path not stop at any gap%6=0 primes.
not stop at 257, not stop at 263,
not stop at 269.
exit path stop at next gap%6=2 prime.
2018-12-03-10-27

<a name=docB113>
2018-12-03-11-07
To choose exit path stop at next gap%6=2
prime, the reason is to simplify initial
study work. In other words
"1FT3" include following pay6 patterns
,1F,3T=gap 10,20 ;
,1F,3T: 1F=1*6+Four=10, 3T=3*6+Two=20
,1F,3T; primes 1627,1637,1657

,1F,3,T=gap 10,18,2
,1F,3,T; primes 1249,1259,1277,1279

<a name=docB114>
,1F,2,1T=gap 10,12,8
,1F,2,1T; primes 1801,1811,1823,1831

,1F,1,1,1T=gap 10,6,6,8
,1F,1,1,1T; primes 1171,1181,1187,1193,1201

,1F,1,1,1,T=gap 10,6,6,6,2
,1F,1,1,1,T; primes 241,251,257,263,269,271

<a name=docB115>
Lump these patterns together as "1FT3" to
simplify initial study.

"1FT3" or "1F3T" the first 1F is 1*6+Four=10.
Left-side pay6 string (entering pay6 string)
1F cannot be splitted to
,1,F,3T //NOT do this
Pay6 string ",1,F,3T" is in ",0F,3T" group.

<a name=docB116>
aa()
"1FT3" or "1F3T" the first 1F is entering pay6
string, second part 3T is leaving pay6 string.
leaving pay6 string has lump sub patterns.
entering pay6 string is clearly identified.
list the summation for every possible
entering plus leaving pay6 string
<a name=hopefail>
with hope //state hope
find one pattern has total zero count of
appearance. After all coding and running
and visual exam the output. All possible
entering plus leaving pay6 string have
matching count much greater than zero.

<a name=docB117>
Prime gap list never show up 2,8 pattern
equivalent situation NOT present in
entering plus leaving pay6 string.

<a name=docB118>
Prime Gap
Six n harmony rule  = avoid_Prime3_cut rule.
demanding the following,
If entering pay6 string contain 'T' then
leaving pay6 string MUST contain 'F' .
If entering pay6 string contain 'F' then
leaving pay6 string MUST contain 'T' .
2018-12-03-12-01

<a name=docB119>
2018-12-03-17-01
Above is lumped (discrete) TF[][], FT[][] RUN j1
RUN j1 , , , show ,
Below is continuous p6tf="" RUN j2
RUN j2 , , , show ,
```<a name=docB120>
button  collect lumped TF[][], and
collect lumped FT[][]. Liu,Hsinhan did not
find empty seats. Then changed attention to
continuous p6tf string. Click
program build continuous p6tf string and not
display result. Click show
Box28 has
[[
0T0_3,0F0_4,0T0_5,0F0_6,0T0_7,0F1_8,0T1_10,
0F0_12,0T0_13,0F2_14,0T1_17,
..... ,
3F1_487,1T4_488,1F0_491,0T0_492,0F1_493,
0T2_495,1F0_497,
]]
<a name=docB121>
aa()
"1T4_488" mean at primeID=488, prime3499
It has 1T4 pattern. Here 1T4=1T4F=1TF4
The prime section is next
3491 gap=8=1*6+Two=1T enter 3499
3499 main prime, gap12=2*6 to 3511
3511 gap6=1*6 to 3517
3517 gap10=1*6+Four to 3527
3527 is "1T4_488" stop point.
3527 is "1F0_491" start point.
```
<a name=docB122>
 From 3491 gap=8=1T entering prime3499 6*582-1=3491 pass gap 1T to 3499=6*583+1 T gap change 6*m-1 prime to 6*n+1 prime from prime3499 leaving gap 4F to prime3527 6*583+1=3499 pass gap 4F to 3527=6*588-1 F gap change 6*m+1 prime to 6*n-1 prime Carefully pay attention to 6*m-1 prime change to 6*n+1 prime or reverse. Main purpose is to avoid prime3 cut. Reduce integer's prime factorization calculation.
```<a name=docB123>
In 1T4=1T4F=1TF4, 4F=(2*6)+(1*6)+(1*6+Four)=28
prime3499 + gap28 = prime3527
Pattern "1T4_488" verified correct.
http://freeman2.com/pay6nxt2.htm#sumRight
Click button  has more.
2018-12-03-18-10

<a name=docB124>
2018-12-03-18-59
Above mentioned T gap, F gap. Now define several
Primes 2,3,5,7,11,13,17,19,23,29,31, .....
Value difference between two neighbor primes
is called prime gap.
Primes 2,3 has gap=1, since 2+1=3.
Primes 3,5 has gap=2, since 3+2=5.
Primes 13,17 has gap=4, since 13+4=17.
Primes 23,29 has gap=6, since 23+6=29.
etc.
<a name=docB125>
Initial primes Primes 2,3 has gap=1 this is
a special case. Only odd number gap. After
2,3 all future primes have even number gap.
prime_gap/6 its remainder has 0,2,4 three
possibility.

<a name=docB126>
aa()
Z gap is prime_gap/6 remainder=0. Example
primes 23,29 has gap=6, 6/6 remainder=0.
primes 199,211,223 has gap=12 twice.
12/6 remainder=0.
primes 523,541 has gap=18, 18/6 remainder=0.
Z in Z gap mean Zero remainder.
Z gap from 6*m+1 prime to 6*n+1 prime, m to n
Z gap from 6*m-1 prime to 6*n-1 prime, m to n
Z gap do not change +1 in 6*m+1 to 6*n+1
Z gap do not change -1 in 6*m-1 to 6*n-1.
<a name=docB127>
Prime gap harmony rule demand
T gap and F gap show up in turn.
Liu,Hsinhan hope find Z gap rule.
6*n-1 side Z gap rule and
6*n+1 side Z gap rule.
see sixn table and pay6 table
In pay6 table
pink q seq. column has T,F,blank;
where blank is omitted Z gap.
Z gap right to blue 1 is 6*n-1 Z gap,
Z gap right to red  1 is 6*n+1 Z gap.
a712091954

<a name=docB128>
T gap is prime_gap/6 remainder=2. Example
primes 5,7 has gap=2, 2/6 remainder=2.
primes 89,97 has gap=8, 8/6 remainder=2.
primes 113,127 has gap=14, 14/6 remainder=2.
T in T gap mean Two remainder.
T gap T=2 change 6*n-1 prime to 6*n+1 prime
T gap T=2 change -1 to +1 and m=n not change.
T gap 1T=8 and greater change 6*m-1 prime
to 6*n+1 prime -1 to +1 and m to n>m.

<a name=docB129>
F gap is prime_gap/6 remainder=4. Example
primes 7,11 has gap=4, 4/6 remainder=4.
primes 139,149 has gap=10,10/6 remainder=4.
primes 1831,1847 has gap=16,16/6 remainder=4.
F in F gap mean Four remainder.
F gap change 6*m+1 prime to 6*n-1 prime
F gap change +1 to -1 and m to n>m.
primes 6*23+1=139 to 149=6*25-1, m=23<25=n ,

<a name=docB130>
pay6 string use one prime as first number,
then follow with pay6 style gaps. Example
2,o,T,T,F,T,F,T,F,1,T,1,F,T,F,1,1,T,1,F,T
second example
59962211,5,1T,2,1,2F,1,T,2,F,1,4,2T,3,1F,2
If a pay6 string without first prime, this
no-prime pay6 string cannot recover prime
array.
<a name=docB131>
aa()
no-prime pay6 string is used to search pay6
string pattern.
One cannot search with-prime pay6 string
223823,1,T,1,F,T,1
One can search no-prime pay6 string
,1,T,1,F,T,1
2018-12-03-19-38

<a name=docB132>
2018-12-05-09-29
pay6get2.htm file size is    562,472 pay6get2.htm
pay6nxt2.htm file size is  8,914,300 pay6nxt2.htm
They have huge difference, pay6nxt2.htm source
code main body is
var pay6million='2,o,T,T,F,T,F,T,F,1, .....
total 8.6 MBytes data
8,636,262 prime.count_3562116_pay6_H.txt
File pay6nxt2.htm can recover prime list from
2 to 60000011 total 3562116 count primes.
<a name=docB133>
goto
http://freeman2.com/pay6nxt2.htm#buildPrimeArr
```
RUN M build , , , , ☜ pID
```<a name=docB134>
click  program build prime array from 2 to
60000011. User may not need display prime list.
Second time click  Small section of prime
list output to Box31.
Small section is determined by numbers in
☜ pID ; pID mean primeID
pID range is 0<= pID <= 3562115
Why restrict pID in the range 0,100000 ?
Why not let pID begin 0 end at 3562115 ?
Explain as next.

<a name=docB135>
pay6nxt2.htm stored prime from 2 to 60000011
total size is 34,867,065 prime.count_3562116.txt
<textarea> box do not support 34.8 MByte size
storage. To get 3562116 primes, each time ask
100000 count primes, repeat do 36 times. Click
value faster.
2018-12-05-10-16

<a name=docB136>
aa()
Below Box31 find following click buttons.
```
ask memory _,_ _,_ _,_ _,_
delete memory array del del del del
QGboxc31.value='' ; delete memory data free memory space.
```<a name=docB137>
2018-12-05-10-24
ask memory  tell you array primeArr[]
current storage. pay6nxt2.htm has only two size
prime storages, one is
primeArr[] size=8 count of primes.
2,3,5,7,11,13,17,19
The other is after click RUN M build
primeArr[] size=3562116 count of primes.

<a name=docB138>
3562116 count of primes use huge memory
Liu,Hsinhan saw one time MSIE auto restart
drop all prime memories. After this event
2018-12-05-10-41

<a name=docB139>
2018-12-05-16-07
Box32 study next gap section. If user input
pay6 pattern [1T,F,T,F,T,F] to
Given pay6 study next gap
then click RUN O1  find pattern
Output to Box32 the following
[[
match list, number is prime (not pID).
T=Two=2, F=Four=4 are prime gap%6
1T=1*6+Two=gap8, 2F=2*6+Four=gap16.
Except initial primes 2,3,5,7 gap T,T
prime gap must be T,F,T,F,T,F in turn.
gap%6=0 not change T,F,T,F pattern.

<a name=docB140>
i0=pID=24,prime=97 ; [1T,F,T,F,T,F]
89,97,101,103,107,109,113:
do.more.here; j0=29, pay6million[j0]=[F] p_new=113
future candidate prime is one of next (tf0=2)
113=113; gap=0, pay6 gap=0 ===== prime
115=5*23; gap=2, pay6 gap=0T
119=7*17; gap=6, pay6 gap=1
121=11*11; gap=8, pay6 gap=1T
125=5*5*5; gap=12, pay6 gap=2
127=127; gap=14, pay6 gap=2T ***** prime

<a name=docB141>
aa()
i0=pID=145804,prime=1954357 ; [1T,F,T,F,T,F]
1954349,1954357,1954361,1954363,1954367,1954369,1954373:
do.more.here; j0=145809, pay6million[j0]=[F] p_new=1954373
future candidate prime is one of next (tf0=2)
1954373=1954373; gap=0, pay6 gap=0 ===== prime
1954375=5*5*5*5*53*59; gap=2, pay6 gap=0T
1954379=7*23*61*199; gap=6, pay6 gap=1
1954381=11*13*79*173; gap=8, pay6 gap=1T
1954385=5*390877; gap=12, pay6 gap=2
1954387=1954387; gap=14, pay6 gap=2T ***** prime

.....

Date/Time: 2018-12-05-16-17-44.753. Elapse:2.26 sec.
<a name=docB142>
gap "1T,F,T,F,T,F" is "8,4,2,4,2,4" From prime2
to prime 60000011 first match is
89,97,101,103,107,109,113:
Main question is what is the next gap after 113?
All possible gaps are 2,4,6,8,10,12 ...

From prime 113, gap=0 is 113 itself.
113=113; gap=0, pay6 gap=0 ===== prime

From prime 113, gap=2 is 115=5*23;
115=5*23; gap=2, pay6 gap=0T=0*6+Two

<a name=docB143>
From prime 113, gap=4 is dropped. Because
between 109,113 gap was 4, by
prime gap harmony rule the following gap
cannot be 4, next gap%6 must be 0 or 2.

From prime 113, gap=6 is 119=7*17;
119=7*17; gap=6, pay6 gap=1=1*6=6

<a name=docB144>
From prime 113, gap=8 is 121=11*11;
121=11*11; gap=8, pay6 gap=1T

From prime 113, gap=10 is 123=3*41;
alert 123=3*41; prime3 show up !!
10%6=4, harmony rule refuse gap=10,
gap=10 no need do integer's prime
factorization. Save calculation time.

From prime 113, gap=12 is 125=5*5*5;
125=5*5*5; gap=12, pay6 record gap12
as 2, because 2*6=12.

<a name=docB145>
From prime 113, gap=14 is 127=127;
127=127; gap=14, pay6 gap=2T ***** prime

"===== prime" indicate it is a prime with
Z gap, gap%6=0
"***** prime" indicate it is a prime with
T gap, gap%6=2 or F gap, gap%6=4
Because between 109,113 gap was 4, Now
stop at gap=14, stop at T gap, 14%6=2

To find future gap need study next one step.
Next two or more steps are not a concern.
2018-12-05-16-48

<a name=docB146>
aa()
If user input pay6 pattern [1T,F,T,F,T,F] to
Given pay6 study next gap
then click RUN O2  find pattern
Output to Box32 the following
[[
match list, number is prime (not pID). T=Two=2, F=Four=4 are prime gap%6
1T=1*6+Two=gap8, 2F=2*6+Four=gap16. Except initial primes 2,3,5,7 gap T,T
prime gap must be T,F,T,F,T,F in turn. gap%6=0 not change T,F,T,F pattern.

<a name=docB147>
i0=pID=24,prime=97 ; [1T,F,T,F,T,F]
89,97,101,103,107,109,113,127,131,137,

i0=pID=145804,prime=1954357 ; [1T,F,T,F,T,F]
1954349,1954357,1954361,1954363,1954367,1954369,1954373,1954387,1954391,1954411,

i0=pID=488538,prime=7187767 ; [1T,F,T,F,T,F]
7187759,7187767,7187771,7187773,7187777,7187779,7187783,7187801,7187857,7187861,

i0=pID=586308,prime=8741137 ; [1T,F,T,F,T,F]
8741129,8741137,8741141,8741143,8741147,8741149,8741153,8741191,8741203,8741231,

.....

Date/Time: 2018-12-05-16-44-02.464. Elapse:2.254 sec.
<a name=docB148>
This  output do not have last gap+2,+4,+6 etc

Both  and  present future data let reader
think why it is so?
Liu,Hsinhan has this utility and data in hand.
Liu did not find out why future data is so? Can you?
Prime puzzle need many people to find out why ?
2018-12-05-17-01

<a name=docB149>
In
http://freeman2.com/pay6nxt2.htm#primeFactorization
has next section
```
from integer to prime factorization ; var prFact; function factInt(arg1,arg2)
```<a name=docB150>
If user input an integer into box, click
Pink background line will show up
2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2 -262144;
Because
262144=2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2
Code display
2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2 -262144;
not display
2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2=262144;
<a name=docB151>
aa()
User can copy and paste
2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2 -262144;
to Real number calculator
http://freeman2.com/pay6nxt2.htm#bx81
2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2 -262144;
get zero answer, confirm integer to prime
factorization success, but output
2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2=262144;
Real number calculator get nothing.
(some freeman2.com web page Real number
calculator auto change '=' to '-' .
pay6nxt2.htm#bx81 has modified version. )
2018-12-05-17-28

<a name=docB152>
Above, click RUN O1  find pattern
Look future one step all gap size integer to
prime factorization result.
Below
http://freeman2.com/pay6nxt2.htm#lookBack
```
<a name=docB153>
from pay6 string look back pay6 pattern
;   ;   Box33 data align right side.
RUN P look back

Box33 ; , , , ☜ pID
```<a name=docB154>
Let user input a pay6 string, program output
look back earlier pay6 strings, let user think
a target pattern and its immediately earlier
neighbor do they have some pattern? For example
twin primes has prime gap=2, in pay6 form it is
",T" Input
Specify pID range  then
Click RUN P look back
Box 33 output the following
<a name=docB155>
[[
3,T,T,F,T,F,T,F,1,T,1,F,T,F,1,1,T,1,F,T,
19,F,1,T,1,F,T,F,1,1,T,1,F,T,1,F,1,1T,F,T,
29,T,1,F,T,F,1,1,T,1,F,T,1,F,1,1T,F,T,F,T,
47,1,1,T,1,F,T,1,F,1,1T,F,T,F,T,F,2T,F,1,T,
59,T,1,F,T,1,F,1,1T,F,T,F,T,F,2T,F,1,T,1F,T,
83,1,1T,F,T,F,T,F,2T,F,1,T,1F,T,1,1,F,1,1,T,
97,F,T,F,T,F,2T,F,1,T,1F,T,1,1,F,1,1,T,1F,T,
103,F,T,F,2T,F,1,T,1F,T,1,1,F,1,1,T,1F,T,F,T,
127,F,1,T,1F,T,1,1,F,1,1,T,1F,T,F,T,2,2,F,T,
139,1F,T,1,1,F,1,1,T,1F,T,F,T,2,2,F,T,F,1,T,
167,1,1,T,1F,T,F,T,2,2,F,T,F,1,T,1F,1,1,1,T,
181,1F,T,F,T,2,2,F,T,F,1,T,1F,1,1,1,T,1,F,T,
199,2,2,F,T,F,1,T,1F,1,1,1,T,1,F,T,1F,2T,F,T,
233,1,T,1F,1,1,1,T,1,F,T,1F,2T,F,T,F,2T,1,1F,T,
307,F,T,F,2T,1,1F,T,F,1,1T,1,1,F,1,1T,F,1T,1F,T,
313,F,2T,1,1F,T,F,1,1T,1,1,F,1,1T,F,1T,1F,T,1F,T,
353,1,1T,1,1,F,1,1T,F,1T,1F,T,1F,T,1,F,1,1T,F,T,
409,1F,T,1F,T,1,F,1,1T,F,T,F,2,1T,F,1T,F,1,2,T,

Date/Time: 2018-12-09-20-24-41.284. Elapse:0.005 sec.
<a name=docB156>
aa()
Re-do pID range, input [ 2000000,2000080 ]
Click RUN P look back
Box 33 output the following
[[
32452591,1,4,1F,3,1T,4,1,6F,5T,3F,1T,8,F,T,4,2F,4T,1F,T,
32452837,F,T,4,2F,4T,1F,T,12F,2T,1F,1,2,7,1,2T,3F,1,3,T,
32453173,3,2,1,7,1,2,3,1,F,4,1,2,2T,3,5,3,F,1,T,
32453431,1F,4T,3F,4,1T,7,1F,1,1T,3F,4,4,3T,5F,2,4T,7,1F,T,

Date/Time: 2018-12-05-18-00-05.567. Elapse:0.032 sec.
Copy one line data, for example, copy
32453173,3,2,1,7,1,2,3,1,F,4,1,2,2T,3,5,3,F,1,T,
paste line to freeman2.com/pay6nxt2.htm#bx22
click RUN g  output to Box23. Make
sure Box23 all number be prime. Easy way to do
click  output to Box23, if no one
number look like 32453193=3*97*229*487 then OK.

<a name=docB157>
Both output every data line right end is ",T" which
represent a twin prime.
Can you find a twin prime earlier neighbor do they
have consistent pattern?
twin prime pay6 string ,T,
cousin prime pay6 string ,F,
sexy prime pay6 string ,1,
2018-12-05-18-03

<a name=docB158>
2018-12-05-19-17
Above is from pay6 string look back pay6 pattern
Below is small prime study.
```
scissors5,7,11,13,17,19 RUN Q
, , ; , ,
; ; ; ; ;
debug ; if check debug box, output large data, slow down.
```<a name=docB159>
Small primes 2,3,5,7,11,13,17,19 enter an
integer's composite component most easily.
Prime 6*n+1 and 6*n-1 structure drop prime2
and drop prime3 automatically. Then
small primes 5,7,11,13,17,19 enter an integer's
composite component easily. User assign
primes to
pattern to
Then click RUN Q
<a name=docB160>
Box34 output next
[[
input pay6 string=[T,F,T,F,4T] Code check small primes 5,7,11,13,17,19
Following exist message number may contain factor 23,29 etc.
search string [Suggest number near] for prime candidate.

n=434118;(6*n-1)/7=372101
n=434119;(6*n-1+2)/5=520943
n=434120;(6*n-1)/13=200363
n=434121;(6*n-1)/5=520945
n=434122 [T,F,T,F,4T] exist. Suggest number near 2604733
But number may contain factor 23,29 etc.

n=434123;(6*n-1+8)/5=520949
n=434124;(6*n-1+2)/5=520949
n=434125;(6*n-1)/7=372107

Date/Time: 2018-12-05-19-51-35.638. Elapse:0.001 sec.
]]
<a name=docB161>
aa()
Output line
n=434118;(6*n-1)/7=372101
say when n=434118, integer (6*n-1)=2604707
has prime7 factor. 2604707=7*233*1597 ;

<a name=docB162>
Output line
n=434122 [T,F,T,F,4T] exist. Suggest number near 2604733
But number may contain factor 23,29 etc.
say when n=434122, integer (6*n-1)=2604731
is a prime. 2604731=2604731*1 ;
Prime2604731 has ,T,F,T,F,4T; gap pattern.

<a name=docB163>
User input
Program exam 5,7,11,13,17,19 only. If (6*n-1)
contain factor 23,29 etc. Code did not detect.
is carefully managed such that n=434122
integer (6*n-1)=2604731 is a prime.
2018-12-05-20-18

<a name=docB164>
In
http://freeman2.com/pay6nxt2.htm#pID_to_prime
Box35 it has
```
Given primeID find prime.
This number is primeID
Box35     ;
```If input a primeID to box, click ,
program output prime number to Box35.
0<=pID<=3562115
2018-12-05-20-40

<a name=docB165>
2018-12-07-17-30
In Box23 input integers (or click  ), click
, program output prime factorization
to Box23. File pay6nxt2.htm main goal is from given
pay6 pattern and find next gap size without doing
integer's prime factorization. But factorization
is still a must step.  is last one
2018-12-07-17-36

aa()

[=][][]
```

<a name=UnicodeSymbol>
UnicodeSymbol : ℂ ℍ ℕ ℙ ℚ ℝ ℤ ℽ ℾ ℿ ⅀ ⅅ ⅆ ⅇ ⅈ ⅉ ; × ÷ ° ◦ º ¹ ² ³
≦ ≠ ≧ ＜ ＝ ＞ ± ≡ ≈ ≌ ≒ ∏ ∑ √ ∛ ∜ ∝ → ∞ ∈ ∀ ∂ ⊥ ∃ ∋ ∆ ∇ ⊿ ∟ ∠ ∫ ∬ ∭ ∮ ∥
≺ ≻ ≼ ≽ ≾ ≿ ⊀ ⊁ ≭ ≮ ≯ ≰ ≱ ≲ ≳ ≴ ≵ ≶ ≷ ≸ ≹ ∧ ∨ ∩ ∪ ∴ ∵ ∶ ∷ ⊂ ⊃ ⊄ ⊅ ⊆ ⊇
Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ ΢ Σ Τ Υ Φ Χ Ψ Ω ☺ ＋ － ＊ ／
α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς σ τ υ φ χ ψ ω ☺ 〈v,w〉 ⊕ ⊙ ○ ● ◎ ■ □ ▢ ▣ ▤ ▥ ▦ ▧ ▨ ▩ ▪ ▫
§ ‰ ¼ ½ ¾ ⅓ ⅔ ⅕ ⅖ ⅗ ⅘ ⅙ ⅚ ⅛ ⅜ ⅝ ⅞ ⅟ ← ↑ → ↓ ↔ ↕ ↖ ↗ ↘ ↙
&#8544;＝Ⅰ &#8545;＝Ⅱ &#8546;＝Ⅲ &#8547;＝Ⅳ &#8548;＝Ⅴ &#8549;＝Ⅵ
&#8550;＝Ⅶ &#8551;＝Ⅷ &#8552;＝Ⅸ &#8553;＝Ⅹ &#8554;＝Ⅺ &#8555;＝Ⅻ
2017-06-20-20-44 Liu,Hsinhan access en.wikipedia.org
https://en.wikipedia.org/wiki/List_of_mathematical_symbols
en.wikipedia.org-wiki-List_of_mathematical_symbols.htm
http://freeman2.com/utility4.htm#extractUnicode
∧,⇒,−,∖,√,≥,≤,∓,⋅,⁄,⇔,∈,ℤ,ℝ,φ,π,∑,∫, ,∮,∯,∰,…,⋯,⋮,⋰,⋱,∴,∵,≠
˜ ,∝,∞,■,□,∎,▮,‣,≈,≃,≅,♎,≒,△,≡,≜,≝,≐,↔,≪,≫,≦,≧,≺,≻,◅,▻,→,⊃,⊆
⊂,∩,ℕ,ℚ,⊇,∪,↦,•,⊧,∨,⊢,⟨,⟩,ψ,α,β,–,‖,⌊,⌉,⌋,⌈,ℂ,∗,∣,∤,∥,∦,⋕,⊥
∃,≀,↯,※,⇐,⊕,⊻,∐,′,∀,∂,∉,∌,∋,ℍ,ω,○,∘,∅,ℙ,†,⊤,—,⊗,⋉,⋊,⋈,​,ℵ,ℶ
δ,∆,⊖,Δ,ƒ,∇,∏,σ,

∑,∫, ,∮,∯=,&#8202;, //a606301442

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

related files

Pay6 Prime number calculator
http://freeman2.com/pay6get2.htm

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http://freeman2.com/pay6nxt2.htm

file name tute0070.htm mean
TUTor, English, 70 th .htm

2018-11-24-16-25 save as tute0070.htm

Pay6 small prime gap data study notes tute0070.htm
The address of this file is
http://freeman2.com/tute0070.htm