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
upload 2018-12-12
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>
 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 
■ ask memory, delete memory 
■ 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>

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> 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> 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 For numerical calculation, please goto 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> 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 For numerical calculation, please goto 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> 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> 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> Not ALL be prime, then Suggest number near 60493 (60493 is prime) is also fake answer. <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>

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>

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>

';' 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>

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>

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> 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> 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> 2018-12-01-09-12 add begin
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. 2018-12-01-09-14 add stop

<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> 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> 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> 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> 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> 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

<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> Below pay6lst0.htm pay6 prime table short section
p seq.
blue:6*n-1
11,59,83 etc.
q seq. gap prime;
red:6*n+1
13,19,43 etc.
pID
prime ID
pay6
string
0o131o
0T252T
0T273T
0F4114F
0T2135T
0F4176F
0T2197T
0F4238F
1 62991
0T23110T
1 637111
0F44112F
0T24313T
0F44714F
1 653151
1 659161
0T26117T
1 667181
0F47119F
0T27320T
1 679211
0F48322F
1 689231
1T897241T
0F410125F
0T210326T
0F410727F
0T210928T
0F411329F
2T14127302T
0F413131F
1 6137321
0T213933T
1F10149341F
0T215135T
1 6157361
1 6163371
0F416738F
1 6173391
1 6179401
0T218141T
1F10191421F
0T219343T
0F419744F

http://freeman2.com/pay6lst0.htm

<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 
pay6get2.htm first upload 2018-11-03 
"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>

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 . 
pay6nxt2.htm first upload 2018-12-12

<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.
<a name=whynxt>
"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

<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>

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>

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>

"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>

"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 
terms used in this page. 
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>

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>

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 
button  and  help you change  
value faster. 
2018-12-05-10-16


<a name=docB136>

In http://freeman2.com/pay6nxt2.htm#askdel
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 
 explain why ask, del ?
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  
ask  get next line 
primeArr[] size=3562116 count of primes.

<a name=docB138>
3562116 count of primes use huge memory 
 and  and  use additional memory. 
Liu,Hsinhan saw one time MSIE auto restart 
drop all prime memories. After this event 
decide write ask/delete section. 
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>

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>

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 
 output three more primes instead. 

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)
factorization answer go here.
<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>

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>

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.
]] <a name=how2verify>
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>

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 
added utility. 
2018-12-07-17-36



[=][][]



<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

Pay6 small prime gap data grouping for study
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
First upload 2018-12-12

Thank you for visiting Freeman's page.
Freeman Liu,Hsinhan 劉鑫漢
2018-12-05-20-42