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-   -   Double Wagstaff prime? (https://www.mersenneforum.org/showthread.php?t=24542)

 sweety439 2019-06-25 20:13

Double Wagstaff prime?

If M(p) = 2^p-1, then M(M(p)) is called [B]double Mersenne number[/B], and if this number is prime, then it is called [B]double Mersenne prime[/B], M(M(p)) is prime for p = 2, 3, 5 and 7, but not for all 11<=p<=59, and the status is unknown for p=61. Now, we consider the Wagstaff number W(p) = (2^p+1)/3 for odd prime p, then W(W(p)) is called [B]double Wagstaff number [/B], and if this number is prime, then it is called [B]double Wagstaff prime[/B], it is known that W(W(p)) is prime for p = 3, 5 and 7, but not for all 11<=p<=29 (the p=23 case is divisible by 129469791307, see [URL="http://factordb.com/index.php?id=1100000000624918997"]factordb[/URL]), but how about p=31 or above? Are there any double Wagstaff primes > W(W(7))? (related to the conjecture that there are no double Mersenne primes > M(M(7)))

 sweety439 2019-06-25 20:17

I know that if M(M(p)) is prime, then M(p) must be itself prime, similarly, if W(W(p)) is prime, then W(p) must be itself prime.

 GP2 2019-06-25 23:15

[QUOTE=sweety439;520084]it is known that W(W(p)) is prime for p = 3, 5 and 7[/QUOTE]

There are no double Wagstaff primes for p ≤ 23. All of these have known factors. For higher p I know of no PRP tests or even the most basic factoring attempts.

[CODE]
[B]3 3[/B] prime
[B]5 11[/B] prime
[B]7 43[/B] prime
[B]11 683[/B] 1676083,26955961001
[B]13 2731[/B] 67399191280564009798331,2252735939855296339250682011
[B]17 43691[/B] 349529
[B]19 174763[/B] 173085275201
[B]23 2796203[/B] 129469791307,36992613766212121
[STRIKE]29 178956971[/STRIKE]
[B]31 715827883[/B]
[STRIKE]37 45812984491[/STRIKE]
[STRIKE]41 733007751851[/STRIKE]
[B]43 2932031007403[/B]
[STRIKE]47 46912496118443[/STRIKE]
[STRIKE]53 3002399751580331[/STRIKE]
[STRIKE]59 192153584101141163[/STRIKE]
[B]61 768614336404564651[/B]
[/CODE]

There are no unknown Wagstaff primes below 10M. Ryan Propper in 2013 searched at least some of the space between 10M and 14M but he does not recall exactly which exponents. I have a hunch that there are no undiscovered Wagstaff primes below 14M.

 R. Gerbicz 2019-06-25 23:59

[QUOTE=GP2;520087]There are no double Wagstaff primes for p ≤ 23. All of these have known factors. For higher p I know of no PRP tests or even the most basic factoring attempts.
[/QUOTE]

If W(p) is composite then W(W(p)) is also composite (and all iterated versions).

 R. Gerbicz 2019-06-26 00:16

W(79) is prime, but W(W(79)) is divisible by 183756724581423634555339057*101874969893105185923314913883, hence it is composite.

ps,
and these has a single google hit: [url]https://www.mersenneforum.org/showpost.php?p=443511&postcount=5[/url]

 axn 2019-06-26 02:40

3290547117383710719111443 | W(W(61))

 GP2 2019-06-26 03:36

W(W(31)) also has a factor. Here's what we have so far:

[CODE]
3 3 prime
5 11 prime
7 43 prime
11 683 1676083,26955961001
13 2731 67399191280564009798331,2252735939855296339250682011
17 43691 349529
19 174763 173085275201
23 2796203 129469791307,36992613766212121
31 715827883 3838477063804290331
43 2932031007403
61 768614336404564651 3290547117383710719111443
79 201487636602438195784363 183756724581423634555339057,101874969893105185923314913883
101
127
167
191
199
313
347
[/CODE]

 mathwiz 2019-06-26 05:22

What program was used to find factors of W(W(61)) onward?

I tried mfaktc compiled with -DWAGSTAFF on W(W(43)) but it seemed to choke on the input...

 sweety439 2019-06-26 06:36

[QUOTE=R. Gerbicz;520090]W(79) is prime, but W(W(79)) is divisible by 183756724581423634555339057*101874969893105185923314913883, hence it is composite.

ps,
and these has a single google hit: [url]https://www.mersenneforum.org/showpost.php?p=443511&postcount=5[/url][/QUOTE]

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 [URL="https://oeis.org/A072226"]OEIS A072226[/URL] = {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
[/CODE]

 R. Gerbicz 2019-06-26 08:38

[QUOTE=GP2;520096]Here's what we have so far:

[CODE]
11 683 1676083,26955961001
[/CODE][/QUOTE]

In this line the cofactor is prime: p189=(2^683+1)/3/1676083/26955961001 .

 GP2 2019-06-26 11:20

[QUOTE=mathwiz;520098]What program was used to find factors of W(W(61)) onward?[/QUOTE]

All Wagstaff factors are 2*k*p + 1 for some k, just like with Mersenne. And k is small enough for those factors that you could quickly find them even with a dumb Python script.

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