Thread: Double Wagstaff prime? View Single Post
2019-06-26, 17:41   #15
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

22×19×41 Posts

Quote:
 Originally Posted by sweety439 I saw this thread and I have a generalization for the double Mersenne numbers, Wagstaff-Mersenne numbers, Mersenne-Wagstaff numbers, double Wagstaff numbers, Mersenne-Fermat numbers and Wagstaff-Fermat numbers, since all the Mersenne numbers, Wagstaff numbers and Fermat numbers are of the form Phi_n(2) for special number n (n is prime, twice an odd prime, or power of 2), I generalize this to general number n: 2^{Phi_n(2)}-1 and (2^{Phi_n(2)}+1)/3 if Phi_n(2) is composite, then both of these two numbers are composite thus we only consider those n such that Phi_n(2) is prime these n are listed in OEIS A072226 = {2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, ...} conjectures: * 2^{Phi_n(2)}-1 is prime only for n = 2, 3, 4, 5, 6, 7, 8, 12 * (2^{Phi_n(2)}+1)/3 is prime only for n = 2, 3, 4, 5, 6, 7, 8, 10, 12, 14 these are Phi_n(2) for n<=128: Code: 1,1 2,3 3,7 4,5 5,31 6,3 7,127 8,17 9,73 10,11 11,2047 12,13 13,8191 14,43 15,151 16,257 17,131071 18,57 19,524287 20,205 21,2359 22,683 23,8388607 24,241 25,1082401 26,2731 27,262657 28,3277 29,536870911 30,331 31,2147483647 32,65537 33,599479 34,43691 35,8727391 36,4033 37,137438953471 38,174763 39,9588151 40,61681 41,2199023255551 42,5419 43,8796093022207 44,838861 45,14709241 46,2796203 47,140737488355327 48,65281 49,4432676798593 50,1016801 51,2454285751 52,13421773 53,9007199254740991 54,261633 55,567767102431 56,15790321 57,39268347319 58,178956971 59,576460752303423487 60,80581 61,2305843009213693951 62,715827883 63,60247241209 64,4294967297 65,145295143558111 66,1397419 67,147573952589676412927 68,3435973837 69,10052678938039 70,24214051 71,2361183241434822606847 72,16773121 73,9444732965739290427391 74,45812984491 75,1065184428001 76,54975581389 77,581283643249112959 78,22366891 79,604462909807314587353087 80,4278255361 81,18014398643699713 82,733007751851 83,9671406556917033397649407 84,20647621 85,9520972806333758431 86,2932031007403 87,41175768098368951 88,1034834473201 89,618970019642690137449562111 90,18837001 91,2380065770834284748671 92,14073748835533 93,658812288653553079 94,46912496118443 95,2437355091657331538911 96,4294901761 97,158456325028528675187087900671 98,4363953127297 99,1010780497307234809 100,1098438933505 101,2535301200456458802993406410751 102,5726579371 103,10141204801825835211973625643007 104,264917625139441 105,473474689919911 106,3002399751580331 107,162259276829213363391578010288127 108,68719214593 109,649037107316853453566312041152511 110,1598509118371 111,2698495133088002829751 112,280379743338241 113,10384593717069655257060992658440191 114,91625794219 115,159734217659271026679184351 116,57646075230342349 117,4140156916495986979321 118,192153584101141163 119,39926307770348782922179133311 120,4562284561 121,1298708349570020393652962442872833 122,768614336404564651 123,690814754065816531725751 124,922337203685477581 125,1267650638007162390353805312001 126,77158673929 127,170141183460469231731687303715884105727 128,18446744073709551617
status of the Wagstaff numbers with these exponents:

Code:
n   Phi_n(2)   known factors of (2^Phi_n(2)+1)/3
1   1 = unit
2   3          prime
3   7          prime
4   5          prime
5   31         prime
6   3          prime
7   127        prime
8   17         prime
9   73         1753,1795918038741070627 (fully factored)
10  11         prime
11  2047 = 23 * 89
12  13         prime
13  8191       (with no known prime factor but definitely composite)
14  43         prime
15  151        18717738334417,50834050824100779677306460621499 (fully factored)
16  257        37239639534523,518144156602508243009,P43 (fully factored)
17  131071     2883563
18  57 = 3 * 19
19  524287     (with no known prime factor but definitely composite)
20  205 = 5 * 41
21  2359 = 7 * 337
22  683        1676083,26955961001,P189 (fully factored)
23  8388607 = 47 * 178481
24  241        2411,10411181203,15059828108442641,P43 (fully factored)
25  1082401 = 601 * 1801
26  2731       67399191280564009798331,2252735939855296339250682011
27  262657     (with no known prime factor but definitely composite)
28  3277 = 29 * 113
29  536870911 = 233 * 1103 * 2089
30  331        5297,2983001129,7520796641,8530674250842274717434530683,P49 (fully factored)
31  2147483647 (with unknown status)
32  65537      13091975735977

Last fiddled with by sweety439 on 2019-06-26 at 17:46