坎宁安第一类素数链

yangchuanju

坎宁安第一类素数链

形如p,2p+1,4p+3,8p+7,……的素数链称之。

yangchuanju

【转载】蔡家雄先生在yangchuanju《连体四生素数》中94楼的问题：
素数链：Q(n+1)=2*Q(n) +1，求证：这条素数链一定是9字尾素数。

素数差(首项差p+1)等比2 的8生素数是存在的。

1------p = 19099919

2------p = 52554569

3------p = 85864769

4------p = 171729539

5------p = 198479579

6------p = 226366139

7------p = 305192579

8------p = 344319029

9------p = 396959159

10------p = 474118349

11------p = 574964129

12------p = 610385159

13------p = 685066979

14------p = 764406059

yangchuanju

尾数皆是9的证明：
对于长度超过4级的坎宁安素数链，若
素数尾数是1，下一级便是3，第3级便是7，第4级便是5，不再是素数，素数链条断裂；
素数尾数是3，下一级便是7，第3级便是5，不再是素数，素数链条断裂；
素数尾数是7，下一级便是5，不再是素数，素数链条断裂；
只有素数尾数是9的，下一级还是9，以后各级尾数都是9。

yangchuanju

2楼转载的蔡家雄问题是一个8级坎宁安素数链，将1-14中的各个素数p分别变成2p+1,4p+3,8p+7,16p+15,32p+31,64p+63,128p+127都是素数；
再往后加下去，都还有许多素数，但不连续了！如
1        19099919 is prime
2        38199839 is prime
3        76399679 is prime
4        152799359 is prime
5        305598719 is prime
6        611197439 is prime
7        1222394879 is prime
8        2444789759 is prime
9        4889579519=11*444507229
10        9779159039=13*19*131*302227
11        19558318079=28537*685367
12        39116636159=43*83*10960111
13        78233272319=5953*13141823
14        156466544639=29*167*32307773
15        312933089279 is prime
16        625866178559=7127*87816217
17        1251732357119 is prime
18        2503464714239 is prime
19        5006929428479=11*249677*1823057
20        10013858856959=33931*295124189
21        20027717713919=2819*7104546901

yangchuanju

已知的8级坎宁安第一类素数链有23条（A138025），首素数分别是：
19099919, 52554569, 85864769, 171729539, 198479579, 226366139, 305192579, 344319029, 396959159, 474118349, 574964129, 610385159, 685066979, 764406059, 1085182229, 1126004219, 1195651139, 1372213439, 1484454269, 1573750049, 2270291519, 2365144079, 2400025739
蔡家雄先生给出的是该素数链中的前14个。

yangchuanju

网页A138030给出9级坎宁安第一类素数链首素数20个：
85864769, 198479579, 305192579, 2400025739, 7606886429, 7755909149, 10378311899, 16744125509, 17029114499, 17117793089, 17574340739, 26089808579, 28889527679, 43390196519, 45424068359, 52179617159, 54293956289, 56838102179, 65639153579, 66655116659
将各个首素数分别变成2p+1,4p+3,8p+7,16p+15,32p+31,64p+63,128p+127,256p+255都是素数。

yangchuanju

坎宁安素数链可能是任意长，目前已知的最大长度是19级。
网页A005602给出前15级坎宁安第一类素数链最小首素数分别是：
13, 3, 41, 509, 2, 89, 1122659, 19099919, 85864769,26089808579, 665043081119, 554688278429, 4090932431513069, 95405042230542329, 90616211958465842219

蔡家雄

素数链：Q(n+1)=3*Q(n) +2，求证：这条素数链一定是9字尾素数。

素数差(首项差2p+2)等比3 的8生素数是存在的，

1------p = 32694619

2------p = 49325849

3------p = 61595309

4------p = 166689599

5------p = 198438029

6------p = 221877809

7------p = 229412819

8------p = 375256979

9------p = 586130089

10------p = 675322129

11------p = 732923729

12------p = 787352719

13------p = 830570999

14------p = 900919949

yangchuanju

记录：
19级首素数2个：
26 digits, 2014-MAR-23, Chermoni/Wroblewski (79910197721667870187016101)
26 digits, 2014-MAR-15, Chermoni/Wroblewski (42008163485623434922152331)

18级首素数2个：
27 digits, 2014-MAR-09, Chermoni/Wroblewski (658189097608811942204322721)
27 digits, 2014-MAR-08, Chermoni/Wroblewski (182927793839529342111307801)

17级首素数7个：
28 digits, 2014-APR-03, Chermoni/Wroblewski (1540797425367761006138858881)
28 digits, 2014-MAR-09, Chermoni/Wroblewski (2*658189097608811942204322721-1)
27 digits, 2014-MAR-08, Chermoni/Wroblewski (465463467291895470464386321)
25 digits, 2008-JUN-10, Wroblewski (1302312696655394336638441)
24 digits, 2008-JUN-05, Wroblewski (127806074555607670094731)
23 digits, 2008-JUN-05, Wroblewski (40244844789379926979141)
22 digits, 2008-MAY-27, Wroblewski (2759832934171386593519)

16级首素数11个：
28 digits, 2014-APR-03, Chermoni/Wroblewski (2*1540797425367761006138858881-1)
28 digits, 2014-MAR-09, Chermoni/Wroblewski (4*658189097608811942204322721-3)
28 digits, 2008-JUN-26, Wroblewski (2368823992523350998418445521)
26 digits, 2008-JUN-12, Wroblewski (20193491108493165642344881)
25 digits, 2008-JUN-10, Wroblewski (2*1302312696655394336638441-1)
24 digits, 2008-JUN-08, Wroblewski (258296136493222766530021)
24 digits, 2008-JUN-05, Wroblewski (2*127806074555607670094731-1)
23 digits, 2008-MAY-29, Wroblewski (91304653283578934559359)
23 digits, 2008-MAY-25, Wroblewski (15745040405007366603151)
21 digits, 2002-FEB, Carmody/Jobling(GenSv) (892390227741617675069)
19 digits, 1997-DEC, Forbes (3203000719597029781)

15级首素数14个：
45 digits, 2016-JAN-01, Balyakin (14354792166345299956567113728*43#-1)
42 digits, 2015-DEC-17, Balyakin (14366587870229772953764096*43#-1)
41 digits, 2014-APR-25, Wroblewski (27353790674175627273118204975428644651729)
32 digits, 2008-JUN-17, Wroblewski (28320350134887132315879689643841)
26 digits, 2008-JUN-12, Wroblewski (71838292723844326926417601)
25 digits, 2008-JUN-10, Wroblewski (5830768257311375388822241)
25 digits, 2008-JUN-05, Wroblewski (2817673877915370357723841)
24 digits, 2008-MAY-29, Wroblewski (662311489517467124375039)
24 digits, 2008-MAY-28, Wroblewski (196426752643710966405599)
24 digits, 2008-MAY-26, Wroblewski (105998574608115401372161)
23 digits, 2008-MAY-25, Wroblewski (2*15745040405007366603151-1)
23 digits, 2004-OCT, Alm/Andersen(Primo) (11993367147962683402919)
22 digits, ??, Forbes (4678714760875524493409)
20 digits, 1997-SEP, Forbes (54269795934721528591)

独舟星海

素数问题是一个开放式宝藏，是一个集宝盆，不会因为你的开采越来越少，而是你的开采使它成级数增长，开的越快，增的越猛，越剧烈，好像宇宙又要爆炸。