Liu,Hsinhan present a thinking problem to general public. 2017-04-16-18-03

to find next prime without integer's

prime multiplication decomposition.

index ; update 2017-05-30

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

below 22=11+11=17+5=19+3; NOT 22=2*11German mathematician C. Goldbach (1690~1764) in his letter addressed to Swiss mathematician L. Euler (1707~1783), Goldbach wrote: Proposition (A) Every even integer ( ≥ 6 ) is the sum of two odd primes; Proposition (B) Every odd integer ( ≥ 9 ) is the sum of three odd primes. They were called Goldbach conjecture. 2016-06-19-09-47 Liu,Hsinhan access and 2016-06-25-15-55 copied https://arxiv.org/ftp/math/papers/0309/0309103.pdf<a name="docA001"> 2016-08-30-09-25 start On 2016-06-23 upload Prime number and prime decomposition http://freeman2.com/prime_e1.htm prime_e1.htm has graph code and drawing board. On 2016-08-06 upload Prime number calculator http://freeman2.com/prime_e2.htm prime_e2.htm has several prime related calculation buttons and in/output boxes. prime_e2.htm deleted graph code and drawing board to save space. <a name="docA002"> On 2016-09-01 upload Six n Prime Table http://freeman2.com/prim6n01.htm ..... http://freeman2.com/prim6n10.htm On 2016-09-01 upload prime number study notes http://freeman2.com/tute0068.htm After build prime_e1.htm , prime_e2.htm , prim6n01.htm to prim6n10.htm Liu,Hsinhan can study prime number, all study notes record in tute0068.htm <a name="docA003"> First, explain Six n Prime Table http://freeman2.com/prim6n01.htm partial table is next

http://freeman2.com/prime_e2.htm | in 6*n±1 | http://freeman2.com/tute0068.htm |

6*n-1 prime | number n | 6*n+1 prime |

5 | 1 | 7 |

11 | 2 | 13 |

17 | 3 | 19 |

23 | 4 | 5^2 |

29 | 5 | 31 |

5^1 * 7^1 | 6 | 37 |

41 | 7 | 43 |

47 | 8 | 7^2 |

53 | 9 | 5^1 * 11^1 |

59 | 10 | 61 |

5^1 * 13^1 | 11 | 67 |

71 | 12 | 73 |

7^1 * 11^1 | 13 | 79 |

83 | 14 | 5^1 * 17^1 |

89 | 15 | 7^1 * 13^1 |

5^1 * 19^1 | 16 | 97 |

101 | 17 | 103 |

107 | 18 | 109 |

113 | 19 | 5^1 * 23^1 |

7^1 * 17^1 | 20 | 11^2 |

2016-08-30-09-40 here <a name="docA004"> First prime is 2, second prime is 3. 2*3=6 Consider 6*n+remainder for n=0 to n=infinity. For n=0 6*n+remainder 0 = 0 6*n+remainder 1 = 1 6*n+remainder 2 = 2 6*n+remainder 3 = 3 6*n+remainder 4 = 4 6*n+remainder 5 = 5 For n=1 6*n+remainder 0 = 6 6*n+remainder 1 = 7 6*n+remainder 2 = 8 6*n+remainder 3 = 9 6*n+remainder 4 =10 6*n+remainder 5 =11 For n=2 6*n+remainder 0 =12 6*n+remainder 1 =13 6*n+remainder 2 =14 6*n+remainder 3 =15 6*n+remainder 4 =16 6*n+remainder 5 =17 etc. <a name="docA005"> For n>0 all 6*n+0, 6*n+2, 6*n+4 are multiple of two, not prime. One exception is n=0 case, 6*0+2=2 is a prime. For n>0 all 6*n+0, 6*n+3 are multiple of three. One exception is n=0 case, 6*0+3=3 is a prime. For general case consideration, drop n=0 special case. Start from n=1, we can say all 6*n+0, 6*n+2, 6*n+4 are multiple of two. all 6*n+0, 6*n+3 are multiple of three. They are not prime for sure. Only 6*n+1, 6*n+5 can be prime. <a name="docA006"> Twin prime is two consecutive primes differ by two. Twin prime is a major concern. Now rewrite 6*n+5 as 6*n+5+6-6=6*(n+1)+5-6=6*m-1. m=n+1 We say 6*n+1, 6*n-1 are possibly prime. Base on above discussion build Six n Prime Table http://freeman2.com/prim6n01.htm In Six n Prime Table No even number, because dropped 6*n, 6*n+2, 6*n+4. No number multiple of three, because dropped 6*n+3. Prime2 and prime3 do not fit 6*n±1. Table start from prime5,7,11. <a name="docA007"> Six n Prime Table has three columns. Middle column is n number, n used in 6*n±1 Middle column is called n column or n line. Left column is 6*n-1 numbers.//CPrimeRight column is 6*n+1 numbers.//BPrimeThese numbers may be prime, may be composite. Six n Prime Table list 6*n-1 prime closer to n line. List 6*n-1 composite far from n line. composite show up as their prime factorization. Similar consideration for 6*n+1 numbers.BPrime CPrime are used in prime_e2.htm Box19 click button [BPrCPr] which build prime table.<a name="docA008"> From n_th prime to n+1_th prime, the difference is n_th prime gap. g[n]=p[n+1]-p[n] Please see Six n Prime Table. From 6*n-1 prime to 6*n+1 prime, gap is 2. Example 5+2=7, 5=6*1-1 and 7=6*1+1. From 6*n-1 prime to 6*n+1 prime, gap is 8=2+6. Example 89+8=97, 89=6*15-1 and 97=6*16+1. From 6*n-1 prime to 6*n+1 prime, gap is 14=2+12. Example 113+14=127, 113=6*19-1 and 127=6*21+1.From 6*n-1 prime to 6*n+1 prime, gap%6=2 is true From 6*n-1 prime to 6*n+1 prime, gap%6=4 is false<a name="docA009"> Above discuss from 6*n-1 prime to 6*n+1 prime. Below discuss from 6*n+1 prime to 6*m-1 prime. Please see Six n Prime Table. From 6*n+1 prime to 6*m-1 prime, gap is 4. Example 19+4=23, 19=6*3+1 and 23=6*4-1. From 6*n+1 prime to 6*m-1 prime, gap is 10=4+6. Example 139+10=149, 139=6*23+1 and 149=6*25-1. From 6*n+1 prime to 6*m-1 prime, gap is 16=4+12. Example 1831+16=1847, 1831=6*305+1, 1847=6*308-1.From 6*n+1 prime to 6*m-1 prime, gap%6=4 is true From 6*n+1 prime to 6*m-1 prime, gap%6=2 is false<a name="docA010">From 6*n-1 to 6*m+1 , MUST have gap%6=2 From 6*n+1 to 6*m-1 , MUST have gap%6=4 This requirement extend to infinity.gap%6=0 is from 6*n-1 to 6*m-1, not cross n line. gap%6=0 is from 6*n+1 to 6*m+1, not cross n line.Gap cross n line must be in turn. If done from 6*n-1 to 6*m+1 , next must be from 6*n+1 to 6*m-1 . If done from 6*n+1 to 6*m-1 , next must be from 6*n-1 to 6*m+1 . This is Six n harmony rule, it is avoid_Prime3_cut rule.<a name="docA011"> youtube video [Proving the Riemann hypothesis 3 of 6] 2:10 say gap [8,6] [6,8] [8,4] [4,8] are allowed gap [8,2] [2,8] [8,8] [4,10] [10,4], [10,10] are NOT allowed. 2:53 say gap [6,6,6] is allowed, but [6,6,6,6] is NOT allowed.Gap sequence 8,2 ; 10,10 ; 2,6,8 etc NEVER SHOW UP, THEY ARE FORBIDDEN.<a name="docA012"> Why gap sequence 2,6,8 not allow? Please access http://freeman2.com/prime_e2.htm#bx11 In gap sequence [d] [ input box ] enter 2,6,6 not enter 2,6,8 because 2,6,8 no match. Click Box11 output match. One answer is 149, 151, 157, 163 Because 2,6,6 add right end 2 get 2,6,8 . You can add right end 2 to 163, see why 165 is not a prime, why gap sequence 2,6,8 no match. Do more experiments, input gap sequence 2,6,10 see why gap sequence 2,6,8 no match. Future add more study notes. Liu,Hsinhan 劉鑫漢 2016-08-30-11-40 <a name="docA013"> update 2017-03-26 2017-03-25-11-07 //please goto tute0067.htm#a603241230 Liu,Hsinhan stopped prime number work about six month. 2017-03-24-11-?? revisit Goldbach EVen Meet PRIme 1 http://freeman2.com/gevmpri1.jpg LiuHH spend half hour find out how to draw gevmpri1.jpg Record key steps to tute0067.htm#a603241230 and add same notes to http://freeman2.com/gevmpri1.jpg The reason suddenly revisit gevmpri1.jpg is that LiuHH pop up a strange thought, whether gevmpri1.jpg contain information allow find future prime without do integer's prime factorization. 2200=2^3*5^2*11^1 negative answer is likely. Positive answer is rare. 2017-03-25-11-22 <a name="docA014"> update 2017-03-30 2017-03-30-11-04 include start graph http://freeman2.com/sixnlaw2.jpg has next document [[Prime Gap Six n harmony rulePrime numbers are those integers which is divisible by itself and one. Example 19=19*1 is prime. 20=2*2*5 is not prime First few primes are 2,3,5,7,11,13,17, 19,23,29,31,37,41,43,47,53,59,61,67.... Formula 6*n-1 and 6*n+1 change n value find all primes. See Six n Prime Tablehttp://freeman2.com/prim6n01.htmLeft is 6*n-1 prime right is 6*n+1 prime Define Prime Gap = PrimeNext - PrimeNow From 5 to 7 (n=1) from 11 to 13 (n=2) the gap is 2=(6*n+1)-(6*n-1)=0+1+1 From 7 to 11, from 13 to 17 need n, n+1 the gap is 4=[6*(n+1)-1]-(6*n+1)=0+6-1-1 Six n Prime Table show that gap 2 cannot follow another gap 2. In other words, red arrow cannot follow other red arrow and gap 4 cannot follow another gap 4. Blue arrow cannot follow another blue.gap 2 must follow gap 4, gap 2, gap 4... Gap sequence 2,2 or 4,4 is not allowed.The reason is simple. Explain as next. <a name="docA015"> Prime 2,3 do not fit 6*n±1 for any n . Start from prime 5,7,11,13,17 ... if p1 and p2 are two neighbor primes p2=p1+2 Integer sequence p1,p1+2,p1+4 one must be divisible by 3. Since p1,p1+2 both be prime must be p1=3*m+2 and p2=3*m+4 3*m+2 , 3*m+4 are not divisible by 3. then p1+4=3*m+6 where 3*m has factor 3, 6 divisible by 3, then p1+4=3*m+6 has factor 3 and p1+4 is not a prime. This conclude red arrow cannot follow another red arrow. Similar reason apply to blue arrow. Black arrow is gap=6 go downward not contribute to "6*n-1 to/from 6*n+1" Purple arrow is gap=8 , 8=6+2 in which 6 no contribution, residual 2 is same as red arrow. Primes 139, 149 has gap=10. 10=6+4 residual 4 is same as blue arrow. Next line help you to understandSix n harmony rule=avoid_Prime3_cut rule2017-03-29-10-30 <a name="docA016">Prime gap sequence 2,8 never occur, because sequence 2,8 violate Six n harmony rule. Prime gap sequence 4,16 never occur, because in 7, 11, 27 ; 7+4=11, 11+16=27=3*9 in 13, 17, 33 ; 13+4=17, 17+16=33=3*11 gap sequence 4,16 third number 27,33 are not primes. 4,16 is same as 4,16%6 same as 4,4http://freeman2.com/prim6n01.htm Six n Prime Table http://freeman2.com/prime_e1.htm Prime drawing board http://freeman2.com/prime_e2.htm Prime calculator http://freeman2.com/tute0067.htm Goldbach conjecture http://freeman2.com/tute0068.htm Prime study notes http://freeman2.com/sixnlaw2.jpg this graph URL 2017-03-29-10-50 Liu,Hsinhan 劉鑫漢 ]] 2017-03-30-11-10 include stop <a name="docA017"> update 2017-04-08 2017-04-08-16-59 include start graph http://freeman2.com/emphtme1.jpg has next document [[Even meet Prime tablehttp://freeman2.com/prime_e3.htm even 20=17+3=13+7 , 20 meet prime 3,7,13,17 , not meet 5,11 x axis: nID=numberID, even=nID*2+6 0,6;1,8;2,10; etc y axis: pID=primeID, pID=0,1,2,3,... prime=2,3,5,7,... Blue=6*n-1 prime=5,11,17. Red=6*n+1 prime=7,13,19 etc Black=prime3. Yellow=even_not_meet prime. Aqua=remote <a name="docA018"> prime gap show up in three locations Any row, yellow is prime gap. No yellow between twinPr From prime3 go up, each row shift right=prime gap If draw prime (not pID) row separate=prime gap scale In prime_e3.htm below [pID row_align] click get left with Horizontal on, click [gevmpr00()] output all row move to left, show every row are same as prime3 http://freeman2.com/prime_e3.htm utility draw this http://freeman2.com/gevmpri0.jpg same graph by VML http://freeman2.com/prime_e1.htm utility draw VML http://freeman2.com/prime_e2.htm prime calculator http://freeman2.com/emphtme1.jpg html table graph 2017-04-07-23-51 Liu,Hsinhan 劉 鑫 漢 ]] <a name="docA019"> 2017-04-08-17-46 start Explain "prime gap show up in three locations" with graph. See next Even meet Prime table. emphtme1.jpgEven Meet Prime Table 01a604112118 change x-axis to even. <a name="docA020"> prime gap first show See prime3 black yellow row. Left most black square is 6=prime3+prime3. 2nd black square is 8=prime3+prime5. 3rd black square is 10=prime3+prime7. 4th yellow square is 12=prime3+number9. 5th black square is 14=prime3+prime11. 1st black to 2nd black has gap2, no yellow. 3rd black to 5th black has gap4, one yellow. 11th black to 14th black has gap6, two yellow. black yellow row carry prime gap information. Similarly blue yellow row carry prime gap information. red yellow row carry prime gap information. <a name="docA021"> prime gap second show from prime03 black yellow row to prime05 blue yellow row relative to prime03 row, prime05 row shift right gap2 displacement. Similarly from prime07 red yellow row to prime11 blue yellow row relative to prime07 row, prime11 row shift right gap4 displacement. Other row shift follow same pattern. <a name="docA022"> prime gap third show This graph y axis use primeID, not use prime. If y axis use prime, it is easy to see from prime03 row to prime05 row has gap2. from prime07 row to prime11 row has gap4. Hope above explanation help you understand "prime gap show up in three locations" 2017-04-08-18-08 <a name=GoldbachConjecture> update 2017-04-16 2017-04-13-18-36 copied from prime_e3.htm#Goldbach

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ prime13, upper red row _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ prime11, upper blue row _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ prime07, lower red row _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ prime05, lower blue row _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ prime03, prime3 black row 03 05 07 __ 11 13 __ 17 19 __ 23 __ __ 29 31 __ __ 37 __ 41 43 __ 47 prime and Gap 06 08 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 even number German mathematician C. Goldbach (1690~1764) in his letter addressed to Swiss mathematician L. Euler (1707~1783), Goldbach wrote: Proposition (A) Every even integer ( ≥ 6 ) is the sum of two odd primes; Proposition (B) Every odd integer ( ≥ 9 ) is the sum of three odd primes. They were called Goldbach conjecture. 2016-06-19-09-47 Liu,Hsinhan access and 2016-06-25-15-55 copied https://arxiv.org/ftp/math/papers/0309/0309103.pdf<a name="docA023"> 2017-04-13-18-40 Goldbach conjecture suggest Every even integer (≥6) is the sum of two odd primes. Each prime add prime3 get an even, for example prime03 + prime3 = even06 prime05 + prime3 = even08 prime07 + prime3 = even10 integ09 + prime3 = even12 prime11 + prime3 = even14 Even 6,8,10,14 two prime decomposition involve 3. Even 12 two prime decomposition not involve 3. Because 12=3+9 and 9 is not a prime and fail "sum of two odd primes" Goldbach statement. But 12=5+7 still satisfy Goldbach conjecture. <a name="docA024"> Following is a graph for Goldbach conjecture in small prime range. <a name="docA025"> Liu,Hsinhan choose nID and pID as coordinate units that is because prime 3,5,7,11,13,17,17,23 ... take more space in y axis direction. Same hight less data. primeID 1,2,3,4,5,6,7,8 ... take less space in y axis direction. Same vertical space more data. nID=0,1,2,3,4,5... match even=6,8,10,12,14,16... General relation is even=nID*2+6 pID=0,1,2,3,4,5... match prime=2,3,5,7,11,13,... Function relation is prime=primeArr(pID) <a name="docA026"> 2017-04-13-19-08 here Please see Even Meet Prime Table 02 prime03 + prime3 = even06 is black/yellow row most left black square A. Below A marked 03,06. This 03 is variable prime03 This 06 is even06 (nID=0 not shown) black square A is constant prime3 Even06, variable prime03 and constant prime3 coincide. <a name="docA027"> prime05 + prime3 = even08 is black/yellow row left black square B. Below B marked 05,08. This 05 is variable prime05, it is blue a0 above "B,05,08 column". This 08 is even08 (nID=1 not shown) black square B is constant prime3 Black B(prime3) + blue a0(prime5) = even8. <a name="docA028"> prime07 + prime3 = even10 is "b0,a1,C,07,10" column. This 07 is variable prime07(RED b0) This 10 is even10 (nID=2 not shown) black square C is constant prime3 Black C(prime3) + RED b0 (prime7) = even10. prime5 + prime5 = even10 this is blue a1. First prime05 and second prime5 coincide. <a name="docA029"> integ09 + prime3 = even12 is "b1,a2,D,09,12" column. 3+9=12 this 09 is not a prime then "blackD" become yellowD. Yellow D is prime3. Integer 9(=3*3) is not in this table 12=7+5 both 7 and 5 are primes, 7,5 show up in "b1,a2,D,09,12" column. Blue a2 is prime5, red b1 is prime7. <a name="docA030">In black/yellow (1)3 row, YELLOW square D alert us that "b1,a2,D,09,12" column not start a new prime row.Because red b1 is second red in "b0,b1,b2,b3,..." row. Red b0 start a new prime row. Red b0 is above BLACK square C, Red b1 not start a new prime row. Red b1 is above YELLOW square D. 2017-04-13-19-55 stop <a name="docA031"> 2017-04-13-22-19 start 14=11+3 is blue c0 + black E 14=7+7 is red b2 + red b2 16=13+3 is red d0 + black F 16=11+5 is blue c1 + blue a4 18=13+5 is red d1 + blue a5 18=11+7 is blue c2 + red b4 20=17+3 is blue e0 + black H 20=13+7 is red d2 + red b5 etc. 2017-04-13-22-26 stop <a name="docA032"> 2017-04-14-15-40 start Goldbach conjecture suggest that even = primeA + primeB ---eq.a01 When we move along x-axis, one variable must change value. In eq.a01 left side even change. In eq.a01 right side two object primeA and primeB. We can set primeB as constant and let primeA change. In black/yellow row primeB=3=constant. In aRow primeB= 5=constant. Even, primeA change. In bRow primeB= 7=constant. same as above. In cRow primeB=11=constant. same as above. In dRow primeB=13=constant. etc. <a name="docA033"> Next explain why black/yellow row all black square form a prime pattern. BlackA: 6=3+3; BlackA=3, BlackA=3 BlackB: 8=5+3; blue a0=5, BlackB=3 BlackC: 10=7+3; red b0=7, BlackC=3 BlackC: 10=5+5; blue a1=5, blue a1=5 YellowD:12=7+5; red b1=7, blue a2=5 YellowD:12=9+3; YellowD=3, 9 is not prime <a name="docA034"> BlackE: 14=11+3; blue c0=11, BlackE=3 BlackE: 14=7+7; red b2=7, red b2=7 BlackF: 16=13+3; red d0=13, BlackF=3 BlackF: 16=11+5; blue a4=5, blue c1=11 YellowG:18=13+5; red d1=13, blue a5=5 YellowG:18=11+7; blue c2=11, red b4=7 YellowG:18=15+3; YellowG=3, 15 is not prime <a name="docA035"> BlackH: 20=17+3; blue e0=17, BlackH=3 BlackH: 20=13+7; red d2=13, red b5=7 BlackI: 22=19+3; red f0=19, BlackI=3 BlackI: 22=17+5; blue e0=17, blue a7=5 BlackI: 22=11+11; blue c4=11, blue c4=11 YellowJ:24=19+5; red f1=19, blue a8=5 YellowJ:24=17+7; blue e2=17, red b7=7 YellowJ:24=13+11; red d4=13, blue c5=11 YellowJ:24=21+3; 21 is not prime <a name="docA036"> BlackK: 26=23+3; blue g0=23, BlackK=3 BlackK: 26=19+7; red f2=19, red b8=7 BlackK: 26=13+13; red d5=13, red d5=13 YellowL:28=23+5; blue g1=23, blue aa=5 YellowL:28=17+11; blue e4=17, blue c7=11 YellowL:28=25+3; 25 is not prime YellowM:30=23+7; blue g2=23, red ba=7 YellowM:30=19+11; red f4=19, blue c8=11 YellowM:30=17+13; blue e5=17, red d7=13 YellowM:30=27+3; 27 is not prime 2017-04-14-16-48 here <a name="docA037"> Please pay attention to above list. All gray background-color primes are even=prime3+NONE_PRIME This NONE_PRIME not match Goldbach conjecture. Thay are yellow square in black/yellow row and out of consideration. Next please pay attention to primes-on-green background-color. Each primes-on-green has a Black#=3. Thay are black squares in Even Meet Prime Table 02. <a name="docA038"> These prime3 has two roles. View horizontally, they are all prime3. View vertically, these black square prime3 match primes 3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61 ... respectively. Because yellow squares represent those odd numbers not match Goldbach conjecture. Above explain black/yellow row, black square has prime number sequential order. How about aRow or "(2)5" row? <a name="docA039"> black/yellow row start from even=6=prime3+prime3 constant is prime3 but in aRow constant is prime5 if in aRow start from even=6=integer1+prime5 integer1 is not a prime, integer1 violate Goldbach conjecture. <a name="docA040"> in aRow start from even=8 =prime3+prime5 next is even=10=prime5+prime5 next is even=12=prime7+prime5 next is even=14=integ9+prime5 next is even=16=prim11+prime5 get same prime number sequential order pattern, only change is constant be prime5 and start from even=8. <a name="docA041"> Similar reason apply to in bRow constant be prime7 and start from even=10. in cRow constant be prime11 and start from even=14. in dRow constant be prime13 and start from even=16. etc. Because prime number has only one sequential, then all rows have identical pattern. In http://freeman2.com/prime_e3.htm#evenMeetPr under "pID rowAlign" change from "normal" to "left" then click RUN ☞ [gevmpr00()] and click DRAW ☞ [show Box22 table] output graph support that in Even Meet Prime Table 02 all rows get prime number sequential order pattern. 2017-04-14-17-33 <a name="docA042"> 2017-04-14-21-08 Even Meet Prime Table 02 is Goldbach conjecture graph. Even Meet Prime Table 03 parallel shift Table 02 all rows to left end. Key point is to see that Table 02 all rows have same structure. Number below black square are primes. Number below yellow columns are non-primes. <a name="docA042"> 2017-04-14-21-15 Next discuss in Even Meet Prime Table 02 each row right shift prime gap distance relative to the row below it. for example jRow j0 j1 j2 j3 j4 j5 (11)37 right shift 6 relative to iRow i0 i1 i2 i3 i4 i5 i6 i7 i8 (10)31 <a name="docA043">Goldbach conjecture say Every even integer ( ≥ 6 ) is the sum of two odd primes even = primeA + primeB ---eq.a02jRow is prime37 row. Let primeB=37 get even = primeA + 37 ---eq.a03 here even and primeA are both variables. Smallest primeA possible is primeA=3. Smallest even possible is even = 3 + 37 = 40 ---eq.a04 Assume (wrong) jRow not right shift 6 relative to iRow Assume (wrong) jRow right shift 4 relative to iRow. Then j0 is above 38, even 38 = 1 + 37 in this relation number 1 is NOT a prime and "38 = 1 + 37" violate Goldbach conjecture. <a name="docA044"> In Even Meet Prime Table prime 3 row start even=3+3=6 prime 5 row start even=5+3=8 3 to 5 gap=5-3=2, then start even 8-6 gap=2 ... prime 37 row start even=37+3=40 prime 41 row start even=41+3=44 37 to 41 gap=41-37=4, then start even 44-40 gap=4 ... prime503 row start even=503+3=506 prime509 row start even=509+3=512 509 to 503 gap=509-503=6, start even 512-506 gap=6 ... This relation let each row right shift prime gap distance relative to the row below it. 2017-04-14-21-38 <a name="docA045"> 2017-04-15-16-43 Even Meet Prime Table 02 has prime columns (black square A,B,B,E,F etc.) and has none prime columns (yellow square D,G,J,L,M etc.) Even Meet Prime Table 04 has prime columns and delete none prime columns. <a name="docA046"> Prime3 column is black A one square. Prime5 column is black B and blue a0. Prime7 column is black C and blue a1, red b0. Prime11 column = black E, yellow a3, red b2 and blue c0. Prime13 column = black F, blue a4, yellow b3, blue c1 and red d0. etc. <a name="docA047"> Start from prime5 up to infinity prime, each prime is either type 6*n-1 or type 6*n+1 . Prime3 do not fit 6*n±1, Prime3 column has just one black square A. Prime5 = 6*1-1, square a0 is blue indicate 5 = type 6*n-1 prime. Prime7 = 6*1+1, square b0 is red indicate 7 = type 6*n+1 prime. Prime11= 6*2-1, square c0 is blue indicate 11= type 6*n-1 prime. Prime13= 6*2+1, square d0 is red indicate 13= type 6*n+1 prime. etc. <a name="docA048"> Please study Even Meet Prime Table 04, look like each prime column top square color match that prime 6*n±1 type, but squares below top and above bottom black square all has opposite color from that prime 6*n±1 type. For example Prime19 has top red f0 square and middle e1, c4, a7 blue squares and botton black I. (yellow d3, b6 both do not satisfy Goldbach conjecture, 22=13+9 and 9 is not a prime) Second example Prime23 has top blue g0 square and middle f2, d5, b8 red squares and botton black K. (yellow e3, c6, a9 all do not satisfy Goldbach conjecture, 26=5+21 and 21 is not a prime) Middle squares color opposite to top square color? Is this accidental or is this a must? 2017-04-15-17-30 <a name="docA049"> 2017-04-15-21-00 The following show that inevenE = primeA + 3 ---eq.a05 if primeA is 6*n-1 type (blue) then other smaller two primes sum to evenE must be 6*n+1 type (red).Rewrite primeA + 3 as evenE = 6*n-1 + 3 = 6*n+2 = 6*(q+r)+2 = (6*q+1) + (6*r+1) ---eq.a06 In primeA + 3 = primeB + primeC ---eq.a07 primeA is 6*n-1 type and primeB, primeC both be 6*n+1 type. Example 29+3=19+13 29=6*5-1 (blue) ; 19=6*3+1 (red) ; 13=6*2+1 (red) <a name="docA050"> Next show that inevenF = primeD + 3 ---eq.a08 if primeD is 6*n+1 type (red) then other smaller two primes sum to evenF must be 6*n-1 type (blue).Rewrite primeD + 3 as evenF = 6*n+1 + 3 = 6*n+4 = 6*n+(6-6)+4 = 6*(n+1)-6+4 = 6*(n+1)-2 = (6*q-1) + (6*r-1) ---eq.a09 In primeD + 3 = primeE + primeF ---eq.a10 primeD is 6*n+1 type and primeE, primeF both be 6*n-1 type. Example 37+3=29+11=23+17 37=6*6+1 (red) 29=6*5-1 (blue) ; 11=6*2-1 (blue) ; 23=6*4-1 (blue) ; 17=6*3-1 (blue) ; 2017-04-15-21-25

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<a name=EMP_to_Prime>Even meet Prime table is2017-04-16-05-28 start build Even Meet Prime Table 05 2017-04-16-06-55 done build Even Meet Prime Table 05 <a name=docA051> 2017-04-16-07-36 Please read above Even Meet Prime Table 05. Purple line is a border line. Left to purple are given primes. Right to purple is future unknown. Task is from primes we have in hand find next prime without integer's prime multiplication decomposition. Although future is unknown, but Even Meet Prime Table allow us know partial future pattern. Right to purple line has fog covered data, they are known. Because Even Meet Prime Table all rows have identical prime sequence. The only difference is each row shift to right a current gap distance. <a name=docA052> Next important property help us is that all prime column has Bottom black square, top blue square, middle all be red square. OR Bottom black square, top red square, middle all be blue square. To find future prime, we can not see future top square be red or blue. But we can see future middle squares. If future middle has red and blue, mixed color tell us this column is not a prime column, stop and exam next gap. If future column has all one color, then Six N Harmony rule determine this one color column is prime or not. Please see first column right to purple line. It is m1 to am column. m1 to am column has all blue color. If m1 to am column is future prime, it has gap 2017-04-16-07-58 think 2017-04-16-08-02 I need more rules to determine next prime. <a name=docA053> 2017-04-16-18-08 Liu,Hsinhan build Even Meet Prime Table 05 and hope to predict next prime without integer's prime multiplication decomposition. This attempt partially achieved, that is In Even Meet Prime Table all rows have same pattern, all be prime sequence, but shift to right at current gap distance. This property give us partial future data, see Table 05 fog covered right half blue-red-yellow squares. <a name=docA054> In Even Meet Prime Table all prime columns have top square match 6*n±1 property to that prime and below top all squares have opposite 6*n±1 property. Determine next prime gap, must refer to current prime gap in hand. For example, if know from 2,3,5,7,11 ... 41,43,47 need predict next gap current prime gap in hand is 47-43=4, then next prime gap size must NOT have nextGap%6=4 This property is governed by Six n harmony rule which is avoid_Prime3_cut rule. <a name=docA055> Liu,Hsinhan stopped at Even Meet Prime Table 05 next gap size determination step. In Table 05 future m1 to z0 column has all blue, but 49 is not a prime. Future m3 to z2 column has all red. m3 to z2 predict next prime 53=47+6 Question is How to drop m1-z0 column in gap size determination step? <a name=docA056> Another example not covered in Table 05 Assume prime 2,3,5,7 ... 113 are in hand next gap is unknown from prime 113 to prime 127 prime, gap, gap%6, 6*n±1 109 , 4 , 4 , 6*n+1 <==all known 113 , 14 , 2 , 6*n-1 <==only 113 known, 14 is big target 127 , 4 , 4 , 6*n+1 <==all unknown (not create such table, see freeman2.com/gevmpri0.jpg) Even Meet Prime Table future fog columns are Data in hand is 109+gap4=113 next (unknown) gap must be gap%6=2 116=113+3 <== 113 known <a name=docA057> Next is five all same color even numbers between prime 113 to prime 127 118=115+3 <== gap=2%6=2 allowed but not prime 122=119+3 <== gap=6%6=0 allowed but not prime 124=121+3 <== gap=8%6=2 allowed but not prime 128=125+3 <== gap=12%6=0 allowed but not prime 130=127+3 <== gap=14%6=2 allowed 127 is prime see http://freeman2.com/gevmpri0.jpgable?to find next prime without integer's prime multiplication decomposition.2017-04-16-17-05 How to choose 130=127+3 from five same color columns? What new rules can be used to drop 118=115+3 drop 122=119+3 drop 124=121+3 drop 128=125+3<a name=docA058> 2017-04-16-18-03 Liu,Hsinhan decide ask general public to think this problem. Many works should be faster than one work. 2017-04-16-18-39 <a name=docA059> update 2017-04-20 2017-04-19-16-10 In Even Meet Prime Table and given data, all prime columns have either top red, middle blue, bottom black or top blue, middle red, bottom black Examples are red, blue, black 7,13,19,31,37,43 etc. red, blue, black is for 6*n+1 type primes. Other examples are blue, red, black 5,11,17,29,41,47 etc. blue, red, black is for 6*n-1 type primes. See Table 05 left half. <a name=docA060> <a name=docA061> In Even Meet Prime Table and FUTURE data, all prime columns top square are unknown, all prime columns bottom square are unknown, FUTURE data prime columns has only middle color. Example, Table 05 right half fog area, square m3 down to z2 is future prime 53. We do not know 53 is prime, then above square m3, we cannot draw a blue square. Below square cL we cannot draw a black square. Compare with none-prime columns, none-prime columns have either whole column red, or whole column blue, or whole column blue AND red. column contain blue AND red are rejected quickly in finding future prime process. But whole column red none-prime columns and whole column blue none-prime columns give us trouble. <a name=docA062> In future fog blue,red,yellow area, these prime columns and none-prime columns look the same, whole column one color, but some is prime column, some not. Even Meet Prime Table 06 show none-prime columns. <a name=docA063> Let graph upper left corner be called "sea" then prime columns and none-prime columns has one point different. prime columns top square TWO sides face sea, none-prime columns top square ONE side face sea. This is obvious in given data, in future fog area, next future row is unknown, we cannot create future row. Fog m1 to mm squares, which square has one side face sea, which square is away from sea that is unclear. <a name=docA064> In Even Meet Prime Table 06 we have all given data. Study given data help us work in future area. Observe Table 06 prime 23 to 29 (gap 6), row 25(28) is all blue. prime 31 to 37 (gap 6), row 35(38) is all red. prime 47 to 53 (gap 6), row 49(52) is all blue. prime 53 to 59 (gap 6), row 55(58) is all blue. prime 61 to 67 (gap 6), row 65(68) is all red. prime 73 to 79 (gap 6), row 77(80) is all red. prime 83 to 89 (gap 6), row 85(88) is all blue. prime 89 to 97 (gap 8), row 91(94) is all blue. prime 89 to 97 (gap 8), row 95(98) is all red. <a name=docA065> Observe 01: all none prime integers 25,49, 55,85,91 have one color blue. In primeM+6=primeN these blue columns are primeM+2 they are 25,49,55,85 In primeM+8=primeN these blue columns are primeM+2 they are 91 (and future not shown) <a name=docA066> Observe 02: all none prime integers 35,65, 77,95 have one color red. In primeM+6=primeN these red columns are primeN-2 they are 35,65,77 In primeM+8=primeN these red columns are primeN-2 they are 95 (and future not shown) <a name=docA067> Observe 03: all none prime integers 9,15,21, 27,33,39,45,51,57,63,69,75,81,87,93 have TWO color red AND blue in one column. These odd numbers are multiple of 3 and gap 6. They are red AND blue in one column, easy to identify and not a concern. <a name=docA068> 2017-04-19-17-21 here primeM+6=primeN, integer primeN-2 is all red. primeM+8=primeN, integer primeN-2 is all red. How about primeM+10=primeN, whether integer primeN-2 is all red? prime 139, 149; primeN-2=147, even=150 prime 181, 191; primeN-2=189, even=192 prime 241, 251; primeN-2=249, even=252 prime 283, 293; primeN-2=291, even=294 prime 337, 347; primeN-2=345, even=348 <a name=docA069> 150=139+11 ; 139=6*23+1 red; 11=6*2-1 blue 150=137+13 150=131+19 150=127+23 150=113+37 150=109+41 150=107+43 150=103+47 150=97+53 150=89+61 150=83+67 150=79+71 note: 150-3=147=3*7*7 , 147 is not a prime get ill 150=139+11 = red+blue 2017-04-19-17-35 <a name=docA070> 2017-04-19-19-03 Next calculate verify [[ Observe 01: all none prime integers 25,49, 55,85,91 have one color blue. In primeM+6=primeN these blue columns are primeM+2 they are 25,49,55,85 ]] <a name=docA071>Assume primeM is 6*n-1 type primeAssume primeM+6=primeN ---eq.a11 oM2=primeM+2 ---eq.a12 //odd M 2 eM2=oM2+3 ---eq.a13 //even M 2 For example primeM=23, primeN=27 oM2=23+2=25 //study odd number 25 eM2=25+3=28 //emp table odd+3=even 28=23+5 //23=6*4-1, 5=6*1-1 28=17+11 //17=6*3-1, 11=6*2-1 <a name=docA072> Assume eM2=primeS+primeT ---eq.a14 Hope find that both primeS and primeT are 6*n-1 type primes. From eq.a13 get eM2=oM2+3 , apply eq.a14 replace eM2 . primeS+primeT=oM2+3 , then oM2=primeS+primeT-3 ---eq.a15 From eq.a12 get oM2=primeM+2 ---eq.a16 <a name=docA073> Both eq.a15 and eq.a16 are oM2, get primeS+primeT-3=primeM+2 ---eq.a17 then primeS+primeT=primeM+5 ---eq.a18 Because primeM is6*n-1type prime //if primeM is 6*n+1 type prime, void following. primeS+primeT=(6*n-1)+5 =6*n+4=6*(n+1)-2 ---eq.a19 Let integer n+1 be p+q primeS+primeT=6*(p+q)-2=(6*p-1)+(6*q-1) ---eq.a20 <a name=docA074>"6*n-1 type prime, (prime+2)+3 all 6*n-1 type" example In Even Meet Prime Table 02, box (g0)23+2=25(g1) even=25+3=28:23,17,11,5 "23,17,11,5" are all 6*n-1 type In Even Meet Prime Table 02, box (h0)29+2=31(h1) even=31+3=34:29,23,17,11,5 "29,23,17,11,5" are all 6*n-1.eq.a20 say if primeM is blue 6*n-1 type prime , then next column odd primeM+2 change to even primeM+5 primeM+5 must be a sum of two 6*n-1 type primes primeS+primeT. Here gapM in primeM+gapM=primeN did NOT enter calculation. Both next column=prime column satisfy or next column=none prime column satisfy. In table 02 prime 23 row (g0,g1,g2,...gc) is blue 6*n-1 type. Square g0 sit on 23 row and 23 column. Next column g1,e4,c7,aa,L belong to above calculation and NOT a prime column. column g1,e4,c7,aa,L has all 6*n-1 (blue) Squares. On top of g1 is blue sea. (on top of g1 is not a red square) Another example, consider prime 29. In table 02 prime 29 row (h0,h1,h2,...h9) is blue 6*n-1 type. Square h0 sit on 29 row and 29 column. Next column h1,g4,e7,ca,ad,O belong to above calculation and IS a prime column. On top of h1 is i0, i0 is new prime 31 after 29. (on top of h1 is not blue sea) 2017-04-19-19-32 <a name=docA075> What happen if primeM is 6*n+1 type prime? eq.a19 change to next primeS+primeT=(6*n+1)+5 =6*n+6=6*(n+1)+0 ---eq.a21 Let integer n+1 be p+q primeS+primeT=6*(p+q)+0=(6*p+1)+(6*q-1) ---eq.a22 eq.a22 say if primeM is red 6*n+1 type prime , then next column odd primeM+2 change to even primeM+5 primeM+5 must be a sum of one 6*n+1 type and one 6*n-1 type primes. Table 02 column i1,h2,g5,...ae,P belong to above calculation. Sum of one 6*n+1 type and one 6*n-1 type that is one red add one blue! Not a prime column for sure. prime column is top red, middle blue bottom black (6*n+1) top blue, middle red bottom black (6*n-1) 2017-04-19-19-57 <a name=docA076> 2017-04-20-00-07 In Table 05 consider three columns m0 L2 k3 ... W greatest known prime 47 m1 L3 k4 ... z0 unknown zone first column. m3 L5 k6 ... z2 unknown zone first prime 53. After 47 we should find 53, but m1 ... z0 column stand on the way to confuse our searching next prime. The following try sum all primes in one column and compare three sums hope be able drop the confusing none prime columns. Assume prime 3,5,7,...23 are given. Refer to Table 02 square labels, Let sumGiven=g0+f2+d5+b8 //drop square K sumFogNP=g1+e4+c7+aa //drop nothing sumFogPr=f5+d8 //drop square h0,N Why drop K(prime3), Why drop h0,N ? Because in fog future h0,N are unknown. Since drop N (prime3) then drop K (3) <a name=docA077> Sum is next. prime 23+3=26 //even 26 = primeA+primeB 26=23+3 //greatest given prime 23 26=19+7 26=13+13 23~~+3~~+19+7+13+13=78 //drop 3(K), since 3(N) unknown sumGiven=23+19+7+13+13=75 //take 75, ignore 78 //after given 23, next future uniform color column 28=23+5 //25 is none prime neighbor 28=17+11 //sumFogNP=sum fog none prime sumFogNP=23+5+17+11=56 //drop nothing //after given 23, next future prime 29 prime 29+3=32 //even 32 = primeA+primeB 32=29+3 //drop square h0,N; 32=19+13 sumFogPr=f5+d8=19+13=32 //drop 29+3 future prime 29 number 29 and 3 are unknown <a name=docA078> sumGiven=75 //prime 23 sumFogNP=56 < 75 sumFogPr=32 < 56 //prime 29Compare data get sumFogPr < sumFogNP < sumGiven prime 23 to prime 29, gap=6. Is this a rule for gap=6 ?<a name=docA079> Assume prime 3,5,7,...29 are given. Future 31 is unknown. sumGiven=column sum of given greatest prime sumFogNP=column sum of future none prime sumFogPr=column sum of future prime prime 29+3=32 //even 32 = primeA+primeB 32=29+3 //greatest given prime 29 32=19+13 sumGiven=29+19+13=61 //prime 29, drop 3 <a name=docA080> 34=31+3 //prime neighbor 34=29+5 34=23+11 34=17+17 sumFogPr=29+5+23+11+17+17=102 //drop 31+3 sumFogPr=102 > 61=sumGivenprime 29 to 31, gap=2, sumFogNP not exist sumFogPr > sumGiven Is this a rule for gap=2 ? Is this a rule for gap%6=2 ?2017-04-20-00-17 <a name=docA081> 2017-04-20-00-47 Assume prime 3,5,7,...41 are given. Refer to Table 02 square labels, Let sumGiven=k0+j2+i5+de+bh //drop square T sumFogPr=k1+h7+ga+ed+aj //drop square L0,U Because in fog future L0,U are unknown. Sum is next prime 41+3=44 //even 44 = primeA+primeB 44=41+3 //greatest given prime 41 44=37+7 44=31+13 41~~+3~~+37+7+31+13=132 //drop 3 sumGiven=41+37+7+31+13=129 <a name=docA082> Assume 43 is future prime. 46=43+3 //prime neighbor 46=41+5 46=29+17 46=23+23~~43+3~~+41+5+29+17+23+23=184 //drop 43+3 sumFogPr=41+5+29+17+23+23=138 > 129 sumFogPr=138 > 129=sumGiven <a name=docA083>prime 41 to 43, gap=2, sumFogNP not exist sumFogPr > sumGiven is this general for gap=2?<a name=docA084> Assume prime 3,5,7,...47 are given. Integer 49 is none prime neighbor. Prime 53 is future prime neighbor. prime 47+3=50 //even 50=primeA+primeB 50=47+3 //greatest given prime 47 50=43+7 50=37+13 50=31+19 sumGiven=47+43+7+37+13+31+19=197 //drop 3 <a name=docA085> 52=47+5 //none prime neighbor 52=41+11 52=29+23 sumFogNP=47+5+41+11+29+23=156 //drop nothing sumFogNP=156<197=sumGiven 56=53+3 //next future prime neighbor 56=43+13 56=37+19 sumFogPr=43+13+37+19=112 //dropped 53+3 sumFogPr=112<197=sumGiven <a name=docA086>Compare above relation, get sumFogPr < sumFogNP < sumGiven prime 47 to prime 53, gap=6 Is this a rule for gap=6 ? Is this a rule for gap%6=0 ?2017-04-20-01-10 done notes 2017-04-20-07-59 done correction <a name=docA087> 2017-04-20-09-37 Above considered gap=2 examples 29 to 31 sumFogPr > sumGiven 41 to 43 sumFogPr > sumGiven gap=6 examples 23 to 29 sumFogPr < sumFogNP < sumGiven 47 to 53 sumFogPr < sumFogNP < sumGiven The following consider gap=4 example. <a name=docA088> Assume prime 3,5,7,...37 are given. Integer 39 is none prime neighbor. Prime 41 is future prime neighbor. prime 37+3=40 //even 40=primeA+primeB 40=37+3 //greatest given prime 37 40=29+11 40=23+17 sumGiven=37+29+11+23+17=117 //drop 3 <a name=docA089> Integer 39 is none prime neighbor. But 39 contain factor 3, skip 39. 41 is next future prime even 44=41+3 //even 44=primeA+primeB 44=41+3 //next future prime 41 is unknown 44=37+7 44=31+13 sumFogPr=37+7+31+13=88 //dropped 41+3 sumFogPr=88<117=sumGiven <a name=docA090>prime 37 to 41, gap=4, sumFogNP not exist sumFogPr < sumGiven is this general for gap=4?<a name=docA091> Assume prime 3,5,7,...43 are given. Integer 45 is none prime neighbor. Prime 47 is future prime neighbor. prime 43+3=46 //even 46=primeA+primeB 46=43+3 //greatest given prime 43 46=41+5 46=29+17 46=23+23 sumGiven=43+41+5+29+17+23+23=181 //drop 3 <a name=docA092> Integer 45 is none prime neighbor. But 45 contain factor 3,5, skip 45. 47 is next future prime even 50=47+3 //even 50=primeA+primeB 50=47+3 //next future prime 47 is unknown 50=43+7 50=37+13 50=31+19 sumFogPr=43+7+37+13+31+19=150 //dropped 47+3 sumFogPr=150<181=sumGiven <a name=docA093>prime 43 to 47, gap=4, sumFogNP not exist sumFogPr < sumGiven is this general for gap=4?<a name=docA094> Summary gap=2 examples 29 to 31 sumFogPr > sumGiven 41 to 43 sumFogPr > sumGiven gap=4 examples 37 to 41 sumFogPr < sumGiven 43 to 47 sumFogPr < sumGiven gap=6 examples 23 to 29 sumFogPr < sumFogNP < sumGiven 47 to 53 sumFogPr < sumFogNP < sumGiven 2017-04-20-10-24 <a name=docA095> 2017-04-20-14-42 Gap 2 primes, 11,13; 17,19; 29,31; 41,43 Gap 4 primes, 7,11; 13,17; 19,23; 37,41; 43,47 Gap 6 primes, 23,29; 31,37; each has special structure. Please open http://freeman2.com/prim6n01.htm and compare prim6n01.htm table top with Even Meet Prime Table 02 below. <a name=docA096> In prim6n01.htm left column is 6*n-1 type prime. In Table 02 blue color squares are 6*n-1 type. In prim6n01.htm right column is 6*n+1 type prime. In Table 02 red color squares are 6*n+1 type. In prim6n01.htm from left to right is from 6*n-1 to 6*n+1 type. In Table 02 blue blue red is from 6*n-1 to 6*n+1. see square label c0,c1,d0; e0,e1,f0; h0,h1,i0; see emp6npe1.jpg graph document. In prim6n01.htm from right to left is from 6*n+1 to 6*n-1 type. In Table 02 red red red blue is from 6*n+1 to 6*n-1. see square b0,b1,b2,c0; d0,d1,d2,e0; f0,f1,f2,g0; <a name=docA097> In prim6n01.htm from left upper to left lower is from 6*n-1 to 6*n-1 type. (type constant) In Table 02 blue blue blue yellow red is from 6*n-1 to 6*n-1. see square g0,g1,g2,g3,h0; Both g0 and h0 are blue (23,29 both be 6*n-1) see emp6npe1.jpg graph document. In prim6n01.htm from right upper to right lower is from 6*n+1 to 6*n+1 type. (type constant) In Table 02 red red red yellow blue is from 6*n+1 to 6*n+1. see square i0,i1,i2,i3,j0; Both i0 and j0 are red (31,37 both be 6*n+1) Type constant two primes must have gap%6=0. That is two primes must have gap=6,12,18,24 ... each gap divide by 6, residual zero. <a name=docA098> In prim6n01.htm from left to right is from 6*n-1 to 6*n+1 type. Immediately again left to right is impossible, because no seats in right side "outer space" and because cut off by prime3. In 11,13,15 third 15 is not a prime. In Table 02 blue blue red is from 6*n-1 to 6*n+1. Never immediately again blue blue red. In Table 02 never see red red up_blue. Because red red up_blue enter "outer space" no seats. see emp6npe1.jpg graph document. <a name=docA099> In prim6n01.htm from right to left is from 6*n+1 to 6*n-1 type. Immediately again right to left is impossible, because no seats in left side "outer space" and because cut off by prime3. In 19,23,27 third 27 is not a prime. In Table 02 red red red blue is 6*n+1 to 6*n-1. Never immediately again red red red blue. In Table 02 never see blue blue blue up_red. Since blue blue blue up_red enter "outer space" no seats there. Although e0,e1,e2,f1 is blue blue blue up_red, but f1 not start a new prime row. Only f0 start a new prime row. see emp6npe1.jpg graph document. 2017-04-20-15-32 <a name=docA100> 2017-04-20-15-52 Previous update 2017-04-16, in Even Meet Prime Table 05 mark Lj,Lk,LL,LM four continuous fog red. This is an error, three continuous red or three continuous blue, this is impossible. Except initial special case, prime 3,5,7 has gap 2,2 all other primes must not have gap 2,2. Four continuous fog red represent gap 2,2,2. Even initial special case do not have this 2,2,2 ! update 2017-04-20 change square LL to yellow. 2017-04-20-15-59 <a name=docA101> 2017-04-20-18-08 One hour ago created http://freeman2.com/emp6npe1.jpg Even Meet Prime Table and Six N Prime Table comparison graph. List graph below 2017-04-20-18-10 <a name=docA102> update 2017-04-23 2017-04-22-17-22 Above sum purple vertical line neighbor column color square value to sumFogPr, sumFogNP and sumGiven. Hope compare sum values to decide find next prime number. The following is another attempt. Different direction but same goal. Even Meet Prime Table 07 is same as Even Meet Prime Table 05 slight change at Table 07 y axis use odd number (Table 05 pID) Table 07 insert a gray vertical column, top mark 47, bottom mark $. Call this $ column. <a name=docA103> <a name=docA104> $ column is copied from given black/yellow row. Black/yellow row has gaps, $ column follows and vertical direction must insert gap rows. Compare $ column with m1 to z0 column, what reason tell us to reject m1 to z0 column (49) as a prime column? what reason tell us to choose m3 to z2 column (53) as next prime column? Liu,Hsinhan did NOT find out reason, Liu upload what is in hand let reader work together. 2017-04-22-17-38 <a name=empt08> <a name=empt09>

<a name=ColumnSum31to61>

column type | odd number | even number | column sum | 6*n-1 drop | 6*n+1 drop | bluPrAll | redPrAll | bluPrSum | redPrSum | columnTop | colBottom |

prime | 31 | 34 | 119 | 0 | 39 | 85 | 70 | 85 | 31 | 31 | 3 |

3*n | 33 | 36 | 144 | 11 | 0 | 85 | 70 | 74 | 70 | 31 | 5 |

odd | 35 | 38 | 57 | 80 | 13 | 80 | 70 | 0 | 57 | 31 | 7 |

prime | 37 | 40 | 80 | 0 | 63 | 80 | 63 | 80 | 0 | 31 | 9 |

3*n | 39 | 42 | 126 | 17 | 0 | 80 | 63 | 63 | 63 | 31 | 11 |

prime | 41 | 44 | 44 | 69 | 19 | 69 | 63 | 0 | 44 | 31 | 13 |

prime | 43 | 46 | 69 | 0 | 50 | 69 | 50 | 69 | 0 | 31 | 15 |

3*n | 45 | 48 | 96 | 23 | 0 | 69 | 50 | 46 | 50 | 31 | 17 |

prime | 47 | 50 | 50 | 52 | 0 | 52 | 50 | 0 | 50 | 31 | 19 |

odd | 49 | 52 | 52 | 0 | 31 | 52 | 31 | 52 | 0 | 31 | 21 |

3*n | 51 | 54 | 54 | 29 | 0 | 52 | 31 | 23 | 31 | 31 | 23 |

prime | 53 | 56 | 0 | 29 | 31 | 29 | 31 | 0 | 0 | 31 | 25 |

odd | 55 | 58 | 29 | 0 | 31 | 29 | 31 | 29 | 0 | 31 | 27 |

3*n | 57 | 60 | 60 | 0 | 0 | 29 | 31 | 29 | 31 | 31 | 29 |

prime | 59 | 62 | 31 | 0 | 0 | 0 | 31 | 0 | 31 | 31 | 31 |

prime | 61 | 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 31 | 33 |

Even Meet Prime analysis 38=31+7=19+19, 38 meet 7,19,31 sum to 57. but 38 not meet 5,7,11,13,17,19,23,29,31.

Even 38 column top=31=given prime, 38 column bottom=7, drop 3,5(<7) because prime 3,5 are not visible.

Primes (3,5,)7,11,13,17,19,23,29,31 < 38-3; Primes >=5 has 6*n-1 and 6*n+1 two types. 11=6*2-1, 19=6*3+1.

6*n-1 primes are 11,17,23,29; bluPrAll=80=11+17+23+29 All four 6*n-1 primes not meet 38, then [6*n-1 drop]=80

6*n+1 primes: 7,13,19,31; redPrAll=70=7+13+19+31; 13(=6*2+1) not meet 38, [6*n+1 drop]=13 (38-13=25 nonPr)

Can you see any reason support 37 be next prime after 31 ? More at http://freeman2.com/tute0068.htm

Liu,Hsinhan 劉鑫漢 build table and document on 2017-04-29-12-00 Table start 2017-04-24-18-50 (a604241850)

<a name=ColumnSumDoc> update 2017-05-03 2017-05-02-15-43 'Column' in 'ColumnSumDoc' mean Columns in Even Meet Prime Table 07 and in Even Meet Prime Table 08 Especially for columns in future fog zone. 'Column' in 'ColumnSumDoc' do NOT mean Columns in Future prime column sum list 2017-05-02-15-49 <a name=docA105> 2017-04-29-17-21 start In http://freeman2.com/prime_e3.htm#bx24 added one section Future prime column sum list table builder For Even Meet Prime Table 08 given prime from 5 to 31, the column sum list table is here In column sum list, first row is given prime 31.prime bgn , endFollowing rows are all future data. The goal is to find whether we can determine next prime 37 without using integer's prime multiplication decomposition. <a name=docA106> Column sum list table structure is the following. The main column is blue/red/silver color column. column type tell current odd number is a prime? or a 3*n number or just an odd number (like 49)

column type odd number even number column sum 6*n-1 drop 6*n+1 drop bluPrAll redPrAll bluPrSum redPrSum columnTop colBottom prime 31 34 119 0 39 85 70 85 31 31 3 Odd number column start from user assigned prime up to user assigned end number. Each row increase 2 to pass even numbers.

column type prime 3*n odd <a name=docA107> Even number column start from user assigned prime add three.

odd number 31 33 35 The reason to add 3 (not add 5,7,11 etc.) that is because all prime (example 31) add prime3 become an even number (example 31+3=34) Prime 31 first time show up at even 34=31+3. Prime 31 not first show up at even 36=31+5. Prime 31 not first show up at even 38=31+7. "add 3" become a guide line. Odd 35 also add 3 get 35+3=38 and 38 is used to do Even Meet Prime analysis for odd 35. (35 is not a prime) <a name=docA108> column sum has red/silver/pink color squares.

even number 34 36 38 red color square indicate this sum for a prime. silver color square indicate this sum for a 3*n. pink color square indicate this sum for an odd. Top first red is from given greatest prime 31. 31+3=34, 34=31+3=29+5=23+11=17+17 34 meet 3,5,11,17,23,29,31 3+5+11+17+23+29+31=119 <a name=docA109> Top second red is from next unknown prime. Is it possible that column sum number sequence reveal next unknown prime? silver color square, for example odd 33 even 36, has higher sum value. If odd is not 3*n then Even Meet Prime analysis has all 6*n-1 type prime (55, 58) or sum has all 6*n+1 type prime (77, 80). But if odd is 3*n then both 6*n-1 type and 6*n+1 type prime enter sum. Sum value is higher. 2017-04-29-18-18 <a name=docA110> column sum is the major column, this column sum what value? Even Meet Prime analysis has two methods. First 40=37+3=29+11=23+17 in http://freeman2.com/prime_e3.htm#bx14 Box enter 40 , choose Click output to Box15 40=37+3 40=29+11 40=23+17 <a name=docA111> Second method in http://freeman2.com/prime_e3.htm#bx24 Enter 37 and 41

column sum 119 144 57 80

Click [evenMeetPrim0()] get nID and pID 17 , 1,4,6,8,9,11 18 , 2,4,5,7,8,9,10,11 19 , 1,3,5,10,11,12 17 is numberID, 17*2+6=even 40 11 is primeID, primeArr[11]=37 click [ID2Pr()] get even and prime 40:3,11,17,23,29,37 42:5,11,13,19,23,29,31,37 44:3,7,13,31,37,41 <a name=docA112> For even 40 (odd 37) the sum is 3+11+17+23+29+37=120 But Future prime column sum list get 11+17+23+29=80 Sum list dropped 37, because given prime 31, future prime 37 is unknown. Sum list dropped 3, because in Table 08 i3,h4 ... z2,37 column do not have prime 3,5,7 Program must drop 3,5,7. Limited future data let even 40 sum 3+11+17+23+29+37=120 cut to 11+17+23+29=80 In Table 08 go further right future, correspond to sum list go further down. In hand data is less and less. 2017-04-29-18-55 <a name=docA113> 2017-05-01-18-17 On 2017-04-30 Liu,Hsinhan added next checkbox Box25 output ;Help ,;unknown future column sum to http://freeman2.com/prime_e3.htm#run2122 If checkbox is unchecked, yellow banner print unknown future column sum If checkbox is checked, yellow banner print ALL GIVEN column sum list The choice "unknown future column sum" is Future prime column sum list table builder original goal. Consider future prime is unknown. Even Meet Prime Table 05 is major concern. <a name=docA114> The choice "ALL GIVEN column sum list" is new added section. Because human already known prime up to 10^22. Within the known range, we are free to set all data be given. To compare future mode answer with ALL GIVEN mode answer it is easy to find out the error in future mode. To add "ALL GIVEN mode" code, the effort is minimum, just modify columnTop and colBottom range.<a name=docA118> update 2017-05-05ALL GIVEN mode [columnTop, colBottom] wider and wider future mode [columnTop, colBottom] narrower and narrower 2017-05-01-18-39 <a name=docA115> 2017-05-02-10-54 Next explain

columnTop colBottom <= left narrower columnTop colBottom 31 3 <= future unknown 31 3 31 5 <= main study 33 3 31 7 right wider => 35 3 31 9 right all given => 37 3 Future prime column sum listcolumnTop constant. prime bgn 31, end 37Prime 31 + 3 = even 34=31+3=29+5=23+11=17+17 34:3,5,11,17,23,29,31 even 34 not meet 7,13,19 34 drop 7+13+19=39. Under [6*n+1 drop] first entry is 39. Odd 33 + 3 = even 36=31+5=29+7=23+13=19+17 36:5,7,13,17,19,23,29,31 even 36 not meet 11 36 drop 11. Under [6*n-1 drop] 2nd entry is 11. <a name=docA116> Wider table is next. (different begin/end)

6*n-1 drop 6*n+1 drop bluPrAll redPrAll bluPrSum redPrSum 0 39 85 70 85 31 11 0 85 70 74 70 80 13 80 70 0 57 0 63 80 63 80 0 All given prime column sum listcolBottom constant prime bgn 5, end 13In this section, [6*n-1 drop] and [6*n+1 drop] are main concern. Follow Goldbach conjecture, carry out Even number two prime sum decomposition. 14=11+3=7+7 ; 14:3,7,11 Even 14 not meet prime5 and 5=6*1-1 Under [6*n-1 drop] column and [prime 11 14 ] row find entry 5. It say Even number two prime sum decomposition dropped 5. Similarly 16=13+3=11+5 ; 16:3,5,11,13 Even 16 not meet 7 , Under [6*n+1 drop] column and [prime 13 16 ] row find entry 7. 2017-05-02-11-39 <a name=docA117> 2017-05-02-14-24 How to get [

column type odd number even number column sum 6*n-1 drop 6*n+1 drop bluPrAll redPrAll bluPrSum redPrSum columnTop colBottom prime 5 8 8 0 0 5 0 5 0 5 3 prime 7 10 15 0 0 5 7 5 7 7 3 3*n 9 12 12 11 0 16 7 5 7 9 3 prime 11 14 21 5 0 16 7 11 7 11 3 prime 13 16 32 0 7 16 20 16 13 13 3 6*n-1 drop] and [6*n+1 drop] ? The calculation is under [bluPrAll] sum all 6*n-1 primes and [redPrAll] sum all 6*n+1 primes followed with calculation [bluPrSum] sum meet 6*n-1 primes and [redPrSum] sum meet 6*n+1 primes Finally all_sum subtract meet_sum get drop_sum. Liu,Hsinhan include [6*n-1 drop] and [6*n+1 drop] hope to get more future data and help to find next gap (future, unknown). 2017-05-02-14-41

2017-05-05-16-42 merge [All given prime column sum list] with [Future prime column sum list] .

Both table (given and unknown) come from http://freeman2.com/prime_e3.htm reader need cut

and paste merge to one table.

Assume primes from 3,5,7 ... 2549,2551,2557 are given. Future are unknown. Goal is to find

next (after 2557) gap size. If find a way to identify next gap size, then next prime=2557+gap

Do you have any good idea? Liu,Hsinhan 2017-05-05-17-01

column type | odd number | even number | column sum | 6*n-1 drop | 6*n+1 drop | bluPrAll | redPrAll | bluPrSum | redPrSum | columnTop | colBottom |

prime1 | 2503 | 2506 | 115276 | 103091 | 204946 | 215861 | 207449 | 112770 | 2503 | 2503 | 3 |

3*n | 2505 | 2508 | 185592 | 122395 | 115323 | 215861 | 207449 | 93466 | 92126 | 2505 | 3 |

odd | 2507 | 2510 | 112950 | 215861 | 94499 | 215861 | 207449 | 0 | 112950 | 2507 | 3 |

odd | 2509 | 2512 | 85408 | 130453 | 207449 | 215861 | 207449 | 85408 | 0 | 2509 | 3 |

3*n | 2511 | 2514 | 181008 | 129125 | 113177 | 215861 | 207449 | 86736 | 94272 | 2511 | 3 |

odd | 2513 | 2516 | 95608 | 215861 | 111841 | 215861 | 207449 | 0 | 95608 | 2513 | 3 |

odd | 2515 | 2518 | 91907 | 123954 | 207449 | 215861 | 207449 | 91907 | 0 | 2515 | 3 |

3*n | 2517 | 2520 | 282240 | 72375 | 68695 | 215861 | 207449 | 143486 | 138754 | 2517 | 3 |

odd | 2519 | 2522 | 88270 | 215861 | 119179 | 215861 | 207449 | 0 | 88270 | 2519 | 3 |

prime1 | 2521 | 2524 | 103484 | 114901 | 207449 | 215861 | 209970 | 100960 | 2521 | 2521 | 3 |

3*n | 2523 | 2526 | 171768 | 130579 | 123484 | 215861 | 209970 | 85282 | 86486 | 2523 | 3 |

odd | 2525 | 2528 | 85952 | 215861 | 124018 | 215861 | 209970 | 0 | 85952 | 2525 | 3 |

odd | 2527 | 2530 | 139150 | 76711 | 209970 | 215861 | 209970 | 139150 | 0 | 2527 | 3 |

3*n | 2529 | 2532 | 177240 | 134229 | 114362 | 215861 | 209970 | 81632 | 95608 | 2529 | 3 |

prime0 | 2531 | 2534 | 103894 | 215861 | 108610 | 218392 | 209970 | 2531 | 101360 | 2531 | 3 |

odd | 2533 | 2536 | 91296 | 127096 | 209970 | 218392 | 209970 | 91296 | 0 | 2533 | 3 |

3*n | 2535 | 2538 | 177660 | 127130 | 123572 | 218392 | 209970 | 91262 | 86398 | 2535 | 3 |

odd | 2537 | 2540 | 109220 | 218392 | 100750 | 218392 | 209970 | 0 | 109220 | 2537 | 3 |

prime1 | 2539 | 2542 | 96596 | 124338 | 209970 | 218392 | 212509 | 94054 | 2539 | 2539 | 3 |

3*n | 2541 | 2544 | 195888 | 119037 | 115976 | 218392 | 212509 | 99355 | 96533 | 2541 | 3 |

prime0 | 2543 | 2546 | 91656 | 218392 | 123399 | 220935 | 212509 | 2543 | 89110 | 2543 | 3 |

odd | 2545 | 2548 | 122304 | 98631 | 212509 | 220935 | 212509 | 122304 | 0 | 2545 | 3 |

3*n | 2547 | 2550 | 252450 | 90534 | 90460 | 220935 | 212509 | 130401 | 122049 | 2547 | 3 |

prime0 | 2549 | 2552 | 109736 | 220935 | 105325 | 223484 | 212509 | 2549 | 107184 | 2549 | 3 |

prime1 | 2551 | 2554 | 103437 | 122601 | 212509 | 223484 | 215060 | 100883 | 2551 | 2551 | 3 |

3*n | 2553 | 2556 | 181476 | 134473 | 122595 | 223484 | 215060 | 89011 | 92465 | 2553 | 3 |

odd | 2555 | 2558 | 78019 | 223484 | 137041 | 223484 | 215060 | 0 | 78019 | 2555 | 3 |

prime1 | 2557 | 2560 | 122880 | 103164 | 215060 | 223484 | 217617 | 120320 | 2557 | 2557 | 3 |

3*n | 2559 | 2562 | 210084 | 123402 | 107615 | 223484 | 217617 | 100082 | 110002 | 2557 | 5 |

3*n | 2559 | 2562 | 210084 | 123402 | 107615 | 223484 | 217617 | 100082 | 110002 | 2559 | 3 |

odd | 2561 | 2564 | 87176 | 223479 | 130441 | 223479 | 217617 | 0 | 87176 | 2557 | 7 |

odd | 2561 | 2564 | 87176 | 223484 | 130441 | 223484 | 217617 | 0 | 87176 | 2561 | 3 |

odd | 2563 | 2566 | 93659 | 129820 | 217610 | 223479 | 217610 | 93659 | 0 | 2557 | 9 |

odd | 2563 | 2566 | 93659 | 129825 | 217617 | 223484 | 217617 | 93659 | 0 | 2563 | 3 |

3*n | 2565 | 2568 | 164352 | 145207 | 131530 | 223479 | 217610 | 78272 | 86080 | 2557 | 11 |

3*n | 2565 | 2568 | 164352 | 145212 | 131537 | 223484 | 217617 | 78272 | 86080 | 2565 | 3 |

odd | 2567 | 2570 | 113080 | 223468 | 104530 | 223468 | 217610 | 0 | 113080 | 2557 | 13 |

odd | 2567 | 2570 | 113080 | 223484 | 104537 | 223484 | 217617 | 0 | 113080 | 2567 | 3 |

odd | 2569 | 2572 | 100308 | 123160 | 217597 | 223468 | 217597 | 100308 | 0 | 2557 | 15 |

odd | 2569 | 2572 | 100308 | 123176 | 217617 | 223484 | 217617 | 100308 | 0 | 2569 | 3 |

3*n | 2571 | 2574 | 221364 | 111462 | 108239 | 223468 | 217597 | 112006 | 109358 | 2557 | 17 |

3*n | 2571 | 2574 | 221364 | 111478 | 108259 | 223484 | 217617 | 112006 | 109358 | 2571 | 3 |

odd | 2573 | 2576 | 110768 | 223451 | 106829 | 223451 | 217597 | 0 | 110768 | 2557 | 19 |

odd | 2573 | 2576 | 110768 | 223484 | 106849 | 223484 | 217617 | 0 | 110768 | 2573 | 3 |

odd | 2575 | 2578 | 91519 | 131932 | 217578 | 223451 | 217578 | 91519 | 0 | 2557 | 21 |

odd | 2575 | 2578 | 91519 | 131965 | 217617 | 223484 | 217617 | 91519 | 0 | 2575 | 3 |

3*n | 2577 | 2580 | 245100 | 101392 | 94537 | 223451 | 217578 | 122059 | 123041 | 2557 | 23 |

3*n | 2577 | 2580 | 245100 | 101425 | 94576 | 223484 | 217617 | 122059 | 123041 | 2577 | 3 |

prime0 | 2579 | 2582 | 89079 | 223428 | 128499 | 223428 | 217578 | 0 | 89079 | 2557 | 25 |

prime0 | 2579 | 2582 | 91661 | 223484 | 128538 | 226063 | 217617 | 2579 | 89079 | 2579 | 3 |

odd | 2581 | 2584 | 108528 | 114900 | 217578 | 223428 | 217578 | 108528 | 0 | 2557 | 27 |

3*n | 2583 | 2586 | 186192 | 129276 | 125538 | 223428 | 217578 | 94152 | 92040 | 2557 | 29 |

odd | 2585 | 2588 | 85404 | 223399 | 132174 | 223399 | 217578 | 0 | 85404 | 2557 | 31 |

odd | 2587 | 2590 | 152810 | 70589 | 217547 | 223399 | 217547 | 152810 | 0 | 2557 | 33 |

3*n | 2589 | 2592 | 173664 | 138122 | 129160 | 223399 | 217547 | 85277 | 88387 | 2557 | 35 |

prime0 | 2591 | 2594 | 86899 | 223399 | 130648 | 223399 | 217547 | 0 | 86899 | 2557 | 37 |

column type | odd number | even number | column sum | 6*n-1 drop | 6*n+1 drop | bluPrAll | redPrAll | bluPrSum | redPrSum | columnTop | colBottom |

This table main purpose to to show you merge two table to one. 2017-05-05-17-05

<a name=docA119> update 2017-05-08 2017-05-06-15-43 if q is a prime, what condition is necessary for (q+2) is also a prime? q%3 >0 , (q+2)%3 >0 q%5 >0 , (q+2)%5 >0 q%7 >0 , (q+2)%7 >0 q%11>0 , (q+2)%11>0 q%13>0 , (q+2)%13>0 q%17>0 , (q+2)%17>0 ..... 2017-05-06-15-56 59%3 2 59%5 4 59%7 3 59%11 4 59%13 7 59%17 8 59%19 2 59%23 13 59%29 1 59%31 28 59%37 22 59%39 20 59%41 18 59%43 16 59%47 12 59%53 6 2017-05-06-15-57 59%3 +2 4 59%5 +2 6 59%7 +2 5 59%11+2 6 59%13+2 9 59%17+2 10 59%19+2 4 59%23+2 15 59%29+2 3 59%31+2 30 59%37+2 24 59%39+2 22 59%41+2 20 59%43+2 18 59%47+2 14 59%53+2 8 <a name=docA120> 2017-05-06-15-59 59%3 +4 6 6%3=0 59+4=63 is not a prime 59%5 +4 8 59%7 +4 7 7%7=0 59+4=63=9*7 is not a prime 59%11+4 8 59%13+4 11 59%17+4 12 12%17=12 59%19+4 6 6%19=6 59%23+4 17 59%29+4 5 59%31+4 32 59%37+4 26 59%39+4 24 24%39=24 59%41+4 22 59%43+4 20 59%47+4 16 59%53+4 10 <a name=docA121> 2017-05-06-16-05 integer's prime multiplication decomposition is reliable. [[ Even meet Prime table is able? ☺ ☼ to find next prime without integer's prime multiplication decomposition. ]] 2017-05-06-16-18 UNable C:\$fm\upload\fm\tute0068.htm [[ Even meet Prime table is UNable? ☺ ☼ to find next prime without integer's prime multiplication decomposition. ]] 2017-05-06-16-24 <a name=docA122> 2017-05-06-17-17 In All given/Future prime column sum list prime0 is 6*n-1 type prime 5,11,17,23 etc. prime1 is 6*n+1 type prime 7,13,19,31 etc. 6*n±1 type prime is a major concern in Even meet Prime table. 2017-05-06-17-22 <a name=docA123> 2017-05-08-11-45 update 2017-05-08 in All given prime column sum list add purple line for given and future border line. In future section (below purple line) inserted blue lines between yellow lines. Blue background lines are from all given prime data, yellow background lines are from future data. Purple line in sum list is same as purple line in Table 08 <a name=docA124> update 2017-05-08 change from Even meet Prime table is able? ☺ ☼ to find ... to Even meet Prime table is UNable? ☺ ☼ to find ... Liu, HsinHan studied even meet Prime table several days, did not find a method to use future data in hand do useful calculation. Add weight to suspicious end. 2017-05-08-12-01 <a name=docA125> 2017-05-30-15-09 update 2017-05-30 Please see Even Meet Prime Table 05 From this table each row is a prime sequence, but shift to right a prime_gap distance. If exam its columns, it has four type different columns. 6*n+1 Prime column, for example 19=6*3+1, 31=6*5+1 etc. Column has red hat, blue cloth, black shoe. 6*n-1 Prime column, for example 23=6*4-1, 41=6*7-1 etc. Column has blue hat, red cloth, black shoe. 3*n odd column, for example 21=3*7, 45=3*15 etc. column has red square/blue square in turn, column has no black shoe. 5*n 7*n odd column, for example 25=5*5, 49=7*7 etc. column has all red or all blue, no hat, no shoe. <a name=docA126> Liu,Hsinhan had a question: If insert more none-prime rows, will change of row alter column characters? In Table 09 added [35 no no] row and added [25 no no] row. In Even Meet Prime Table all red row are 6*n+1 primes, all blue row are 6*n-1 primes. Because 35=6*6-1 [35 no no] row color blue. Because 25=6*4+1 [25 no no] row color red. The result is Even Meet Prime Table 09. This table show that If insert more none-prime rows, this insertion will not change column characters. <a name=docA127> Study Even Meet Prime Table 07 and Table 09 after maximum given prime 47, the determination of next prime cannot rely on column pattern, the determination of next prime 53 can be done by integer's prime multiplication decomposition reject 49=7*7, reject 51=3*17 . Can you find next prime gap from future data in hand? Can you find prime rules from future data in hand? Liu,Hsinhan 劉鑫漢 2017-05-30-15-51 <a name=docA128> 2017-05-30-17-32 on 2017-05-30-16-16 get the idea study Even Meet Prime Table 07 column pattern see what information it carry. Build Even Meet Prime Table 10a Even Meet Prime Table 10b Even Meet Prime Table 10c Difference is the purple column mark. <a name=empt10> <a name=empt10b> <a name=empt10c> 2017-05-30-17-35 stop [=][][]

<a name="bx34">

Box34 input ; replace string utility, general but no newline. Click [Box34] [replaceAA]

from to ; from to

from to ; from to

Box35 output ; example to ,

Box36 debug ;

QDboxc36.value='' ;

<a name="docA999"> x_{0}+i*y_{0}x_{0}+∆x u_{x}=v_{y}ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ ┌│┐│ ΪΫάέήίΰ αβγδεζηθικλμνξοπρςστυφχψω <a name="NumberSetsChar"> ℂ Complex numbers ; 複數 ℍ Hello ; ℕ Natural numbers ; 自然數（正整數及零） ℙ Prime numbers ; 素數 ℚ Quotient, Rational numbers ; 有理數 ℝ Real numbers ; 實數 ℤ Zahl, Integers ; (from Zahl, German for integer) ; ℤ 整數（正整數及零及負整數） 2015-03-13-18-52

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Prime number study notes

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