Double Wagstaff prime?

[GP2]

Originally Posted by R. Gerbicz

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

Yes. The smallest Wagstaff number that is not fully factored has exponent 1063.

[GP2]

Originally Posted by mathwiz

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

Originally Posted by GP2

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.

The Double Mersenne Prime Search uses a program called mmff.exe, which is derived from mfaktc.exe

As you mentioned, with mfaktc.exe it suffices to set the -DWAGSTAFF flag to make it find Wagstaff factors instead of Mersenne factors. So maybe that will work with mmff.exe as well, and it might be possible to find a large-ish factor for W(W(43)).

Edit: from looking at the source code, it's not that simple.

[Dylan14]

Originally Posted by GP2

The Double Mersenne Prime Search uses a program called mmff.exe, which is derived from mfaktc.exe

As you mentioned, with mfaktc.exe it suffices to set the -DWAGSTAFF flag to make it find Wagstaff factors instead of Mersenne factors. So maybe that will work with mmff.exe as well, and it might be possible to find a large-ish factor for W(W(43)).

Edit: from looking at the source code, it's not that simple.

Instead of trying to modify mmff to handle double Wagstaff numbers, it might be easier to modify dmdsieve from the mtsieve suite. In this case the appropriate header to pass to pfgw would be

ABCD 2*$a*((2^p+1)/3)+1

where $a is a k value not sieved out and p is an exponent of a Wagstaff (probable) prime.
(Of course, more work is needed to get the sieve to work.)

[sweety439]

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:

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:

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

[R. Gerbicz]

The common generalization could be:

a(n)=polcyclo(h*n,2) for fixed h>0 integer.

For h=1 a(a(p))=M(M(p)) if p and M(p)=2^p-1 is prime.
For h=2 a(a(p))=W(W(p)) if p and W(p)=(2^p+1)/3 is prime.
And you can see this for h>2 also.

Or you can even drop the n=p requirement (ofcourse in this case a(n)!=M(n) for h=1 etc.),
note that we can see a(n)=prime or a(a(n))=prime for composite n values also.

[sweety439]

Originally Posted by R. Gerbicz

The common generalization could be:

a(n)=polcyclo(h*n,2) for fixed h>0 integer.

For h=1 a(a(p))=M(M(p)) if p and M(p)=2^p-1 is prime.
For h=2 a(a(p))=W(W(p)) if p and W(p)=(2^p+1)/3 is prime.
And you can see this for h>2 also.

Or you can even drop the n=p requirement (ofcourse in this case a(n)!=M(n) for h=1 etc.),
note that we can see a(n)=prime or a(a(n))=prime for composite n values also.

The common generalization is Phi(Phi(n,2),2) and Phi(2*Phi(n,2),2), with any integer n

However, there are no known n such that Phi(n,2) is composite but Phi(Phi(n,2),2) is prime, also no known n such that Phi(n,2) is composite but Phi(2*Phi(n,2),2) is prime

[R. Gerbicz]

Originally Posted by sweety439

The common generalization is Phi(Phi(n,2),2) and Phi(2*Phi(n,2),2), with any integer n

But why are you repeating me? You have even an error in this line. Yeah, copying isn't that easy.

[Batalov]

Shhhhhh.... How dare you! You are arguing with the inventor of the Double Wagstaff primes. No one thought of that before, and now someone finally has.

{/sarcasm}

[rogue]

Originally Posted by Dylan14

Instead of trying to modify mmff to handle double Wagstaff numbers, it might be easier to modify dmdsieve from the mtsieve suite. In this case the appropriate header to pass to pfgw would be

ABCD 2*$a*((2^p+1)/3)+1

where $a is a k value not sieved out and p is an exponent of a Wagstaff (probable) prime.
(Of course, more work is needed to get the sieve to work.)

mmff is far faster than dmdsieve for smaller exponents. I don't know if it has a limit for larger exponents. If it does, then it probably wouldn't be difficult to create a "dwdsieve".

[lalera]

hi,
mmff does not (yet) run with nvidia turing cards

[Dylan14]

Originally Posted by rogue

mmff is far faster than dmdsieve for smaller exponents. I don't know if it has a limit for larger exponents. If it does, then it probably wouldn't be difficult to create a "dwdsieve".

Looking at the source code for mmff, it appears the limits are
Exponents 31, 61, 89, 107, 127 for double Mersennes and
Exponents <= 223 for Fermat numbers.
Largest bit level it can test depends on the exponent (see lines 356-455 of mfaktc.c in the source of mmff).