A Sierpinski/Riesel-like problem

[sweety439]

Originally Posted by sweety439

I tested some bases b>64.

S65 has CK=10 (proven)
Covering set: {3, 11}

k,n
1,2
2,1
3,2
4,2
5,1
6,5
7,2
8,1
9,1

S67 has CK=26 (3 k's remain: 1, 17 and 21)
Covering set: {3, 7, 31}

k,n
1,???
2,6
3,1
4,1
5,6
6,4532 (given by CRUS)
7,135
8,1
9,2
10,1
11,209
12,135
13,2
14,1
15,1
16,3
17,???
18,2
19,21
20,2
21,???
22,3
23,1
24,1
25,2

S68 has CK=22 (2 k's remain: 1 and 17)
Covering set: {3, 23}

k,n
1,??? (at n=2^24-1, see GFN stats)
2,1
3,2
4,6
5,29
6,1
7,2
8,319
9,1
10,6
11,3947 (given by CRUS)
12,656921 (given by CRUS)
13,26
14,1
15,1
16,36
17,??? (at n=1M, see CRUS)
18,2
19,6
20,1
21,1

S69 has CK=6 (proven)
Covering set: {5, 7}

k,n
1,2
2,1
3,2
4,1
5,1

S71 has CK=5 (proven)
Covering set: {2, 3}

k,n
1,2
2,3
3,1
4,22

S73 has CK=47 (2 k's remain: 14 and 21)
Covering set: {2, 5, 13}

k,n
1,1
2,4
3,4
4,1
5,1
6,1
7,2
8,28
9,2
10,3
11,1
12,1
13,23
14,???
15,1
16,40
17,9
18,2
19,1
20,1
21,???
22,1
23,2
24,1
25,10
26,1
27,4
28,2
29,1
30,2
31,1
32,2
33,6
34,3
35,1
36,7
37,6
38,6
39,350
40,3
41,1
42,1
43,2
44,2
45,4
46,1

S74 has CK=4 (proven)
Covering set: {3, 5}

k,n
1,2
2,1
3,1

S75 has CK=37 (1 k remain: 11)
Covering set: {2, 19}

k,n
1,32
2,1
3,1
4,2
5,48
6,2
7,1
8,1
9,6
10,1
11,???
12,57
13,2
14,1
15,1
16,1
17,128
18,57
19,3
20,2
21,2
22,4
23,1
24,1
25,2
26,1
27,1
28,129
29,2
30,1
31,1
32,2
33,18
34,1
35,11
36,1

This is the text file for extended Sierpinski problem base 65 to 105. (except bases 66, 70, 78, 82 and 96) (bases 72, 80 and 102 only for the k's not in CRUS, i.e. gcd(k+1,b-1) is not 1)

Note:

For S81, all k = 4*m^4 proven composite by full algebra factors.

The prime 6*67^4532+1 (S67, k=6) is given by CRUS.
The prime 11*68^3947+1 (S68, k=11) is given by CRUS.
The prime 12*68^656921+1 (S68, k=12) is given by CRUS.
The prime 4*77^6098+1 (S77, k=4) is given by CRUS.
The prime (41*81^1223+1)/2 (S81, k=41) is converted by S3, k=41.
The prime 558*81^51992+1 (S81, k=558) is given by CRUS.
The prime 4*83^5870+1 (S83, k=4) is given by CRUS.
The prime 12*87^1214+1 (S87, k=12) is given by CRUS.
The prime 2*101^192275+1 (S101, k=2) is given by CRUS.
The prime 2*104^1233+1 (S104, k=2) is given by CRUS.

Some test limits converted by CRUS:

S68, k=17: at n=1M.
S80, all remain k != 78 mod 79: at n=250K.
S86, k=8: at n=1M.
S93, k=62: at n=100K.
S97, k=120: at n=100K.
S102, all remain k != 100 mod 101: at n=250K.

Some test limits converted by GFN stats:

S68, k=1: at n=2^24-1.
S72, k=72: at n=2^24-2.
S86, k=1: at n=2^24-1.
S92, k=1: at n=2^24-1.
S98, k=1: at n=2^24-1.
S104, k=1: at n=2^24-1.

[sweety439]

Originally Posted by sweety439

R65 has CK=10 (proven)
Covering set: {3, 11}

k,n
1,19
2,4
3,1
4,9
5,2
6,1
7,1
8,10
9,1

R67 has CK=33 (1 k remain: 25)
Covering set: {2, 17}

k,n
1,19
2,768 (given by CRUS)
3,2
4,1
5,1
6,1
7,2
8,2
9,3
10,1
11,6
12,1
13,7
14,1
15,4
16,(proven composite by partial algebra factors)
17,1
18,7
19,8
20,2
21,27
22,1
23,42
24,1
25,???
26,1
27,2
28,2
29,1
30,2
31,10
32,1

R68 has CK=22 (proven)
Covering set: {3, 23}

k,n
1,5
2,4
3,10
4,1
5,13574 (given by CRUS)
6,2
7,25395 (given by CRUS)
8,62
9,3
10,53
11,198
12,2
13,1
14,4
15,1
16,1
17,2
18,1
19,1
20,2
21,1

R69 has CK=6 (proven)
Covering set: {5, 7}

k,n
1,3
2,1
3,1
4,(proven composite by partial algebra factors)
5,4

R71 has CK=5 (proven)
Covering set: {2, 3}

k,n
1,3
2,52
3,2
4,1

R73 has CK=112 (2 k's remain: 79 and 101)
Covering set: {5, 13, 37}

k,n
1,5
2,2
3,1
4,1
5,2
6,2
7,2
8,8
9,(proven composite by partial algebra factors)
10,3
11,1
12,11
13,1
14,1
15,1
16,1
17,15
18,4
19,3
20,1
21,1
22,2
23,1
24,3
25,(proven composite by partial algebra factors)
26,50
27,2
28,1
29,3
30,2
31,3
32,24
33,5
34,1
35,1
36,(proven composite by partial algebra factors)
37,2
38,9
39,1
40,5
41,6
42,50
43,1
44,12
45,1
46,1
47,2
48,73
49,1
50,2
51,1
52,2
53,1
54,63
55,1
56,6
57,4
58,25
59,1
60,9
61,39
62,8
63,2
64,5
65,1
66,1
67,3
68,2
69,1
70,2
71,1
72,8
73,4
74,3
75,5
76,18
77,8
78,1
79,???
80,1
81,1
82,4
83,26
84,1
85,2
86,1
87,3
88,1
89,32
90,1
91,3
92,2
93,1
94,1
95,1
96,2
97,47
98,4
99,1
100,1
101,???
102,10
103,5
104,1
105,102
106,1
107,2
108,1
109,4
110,2
111,1

R74 has CK=4 (proven)
Covering set: {3, 5}

k,n
1,5
2,132
3,2

R75 has CK=37 (1 k remain: 35)
Covering set: {2, 19}

k,n
1,3
2,1
3,16
4,5
5,9
6,1
7,2
8,1
9,1
10,2
11,2
12,2
13,1
14,1
15,2
16,119
17,5
18,54
19,2
20,1
21,1
22,15
23,4
24,2
25,1
26,1
27,2
28,1
29,1
30,41
31,2
32,1
33,1
34,1
35,???
36,1

This is the text file for extended Riesel problem base 65 to 105. (except bases 66, 70, 78, 82, 88 and 96) (bases 72, 80 and 102 only for the k's not in CRUS, i.e. gcd(k-1,b-1) is not 1)

Note:

For R67, all k = m^2 with m = 4 or 13 mod 17 proven composite by partial algebra factors.
For R69, all k = m^2 with m = 2 or 3 mod 5 proven composite by partial algebra factors.
For R73, all k = m^2 with m = 6 or 31 mod 37 and all k = m^2 with m = 3 or 5 mod 8 proven composite by partial algebra factors.
For R79, all k = m^2 with m = 2 or 3 mod 5 proven composite by partial algebra factors.
For R81, all k = m^2 proven composite by full algebra factors.
For R84, all k = m^2 with m = 2 or 3 mod 5 proven composite by partial algebra factors.
For R90, all k = m^2 with m = 5 or 8 mod 13 proven composite by partial algebra factors.
For R94, all k = m^2 with m = 2 or 3 mod 5 proven composite by partial algebra factors.
For R99, all k = m^2 with m = 2 or 3 mod 5 proven composite by partial algebra factors.
For R100, all k = m^2 proven composite by full algebra factors.
For R105, all k = m^2 with m = 3 or 5 mod 8 proven composite by partial algebra factors.

The prime 2*67^768-1 (R67, k=2) is given by CRUS.
The prime 5*68^13574-1 (R68, k=5) is given by CRUS.
The prime 7*68^25395-1 (R68, k=7) is given by CRUS.
The (probable) prime (1*91^4421-1)/90 (R91, k=1) is given by http://oeis.org/A084740 (or http://www.fermatquotient.com/PrimSerien/GenRepu.txt).
The prime 74*100^44709-1 (R100, k=74) is given by CRUS.

Some test limits converted by CRUS:

R80, all remain k != 1 mod 79: at n=250K.
R94, k=29: at n=1M.
R97, k=8: at n=100K.
R102, all remain k != 1 mod 101: at n=200K.

[sweety439]

This is the text file for the conjectured k for all extended Sierpinski/Riesel bases 2<=b<=128, except 66, 120 and 126.

[sweety439]

Originally Posted by sweety439

This is the text file for the conjectured k for all extended Sierpinski/Riesel bases 2<=b<=128, except 66, 120 and 126.

Fixed the text file (type error).

[sweety439]

Update the text file to include the conjectured k for SR126.

Now, all conjectured k for all Sierpinski/Riesel bases 2<=b<=128 were found except SR66 and SR120.

[sweety439]

Originally Posted by sweety439

These text files are the status for these conjectures: (only for bases 5, 8, 9, 11, 13, 14 and 16) (the prime (41*9^2446+1)/2 (S9, k=41) is converted by S3, k=41) (the prime 186*16^5229+1 (S16, k=186) is converted by S4, k=186)

These bases and k's are remain:

base    k
S13     29
S16     89, 215, 459, 515

Interestingly, there are no k's remain in the Riesel conjectures.

Thus, for all these Sierpinski/Riesel bases except S13 and S16, all of the 1st, 2nd and 3rd conjectures are proven. (the 1st and 2nd conjectures for S13, and the 1st conjecture for S16 are also proven, but the 3rd conjecture for S13, and the 2nd and 3rd conjecture for S16 are not proven)

These are the text files for the 1st, 2nd and 3rd conjectures for SR4, all of these conjectures are proven. (the only non-certified PRP in these conjectures is (751*4^6615-1)/3) (the prime 766*4^3196+1 (S4, k=766) is converted by S2, k=383) (the prime 74*4^1276-1 (R4, k=74) is converted by R2, k=74) (the prime 659*4^400258-1 (R4, k=659) is converted by R2, k=659) (the prime 674*4^5838-1 (R4, k=674) is converted by R2, k=674) (the prime 1103*4^2203-1 (R4, k=1103) is converted by R2, k=1103)

[sweety439]

These are the text files for R70 and R88, tested to n=1000.

[sweety439]

Originally Posted by sweety439

These are the text files for R70 and R88, tested to n=1000.

R88, k=400:

for even n let n=2*q; factors to:
(20*88^q - 1) *
(20*88^q + 1)
odd n:
covering set 3, 7, 13

thus proven composite by partial algebra factors.

[sweety439]

The remain k's for these Sierpinski bases are:

base   remain k
S65    none (proven)
S67    1, 17, 21
S68    1, 17
S69    none (proven)
S71    none (proven)
S72    72
S73    14, 21
S74    none (proven)
S75    11
S76    none (proven)
S77    1
S79    none (proven)
S80    86, 92, 166, 295, 326, 370, 393, 472, 
       556, 623, 628, 692, 778, 818, 947, 968
S81    34, 75, 239, 284, 311, 317, 335, 389, 
       439, 514, 569
S83    1, 3
S84    none (proven)
S85    70
S86    1, 8
S87    none (proven)
S88    8
S89    1
S90    none (proven)
S91    1
S92    1
S93    19, 36, 43, 62, 67, 87, 93
S94    none (proven)
S95    none (proven)
S97    1, 22, 27, 43, 62, 64, 83, 97, 116, 120, 123
S98    1
S99    1
S100   none (proven)
S101   none (proven)
S102   122, 178, 236
S103   7, 13
S104   1
S105   36, 191

[sweety439]

The remain k's for these Riesel bases are:

base   remain k
R65    none (proven)
R67    25
R68    none (proven)
R69    none (proven)
R70    278, 376, 434, 489, 496, 729, 811
R71    none (proven)
R72    none (proven)
R73    79, 101
R74    none (proven)
R75    35
R76    none (proven)
R77    none (proven)
R79    none (proven)
R80    10, 31, 214
R81    none (proven)
R83    none (proven)
R84    none (proven)
R85    61, 64, 169
R86    none (proven)
R87    none (proven)
R88    17, 46, 49, 68, 79, 89, 94, 179, 212, 235,
       277, 346, 380, 444, 464, 477, 508, 522,
       536, 541, 544
R89    none (proven)
R90    none (proven)
R91    27
R92    none (proven)
R93    33, 69, 109, 113, 125, 149, 177
R94    16, 29
R95    none (proven)
R97    8, 16, 22
R98    none (proven)
R99    none (proven)
R100   133
R101   none (proven)
R102   191, 207, 1082, 1369
R103   none (proven)
R104   none (proven)
R105   73, 137, 148, 265

[sweety439]

Reserve all 1k base (only 1 remain k that is neither GFN nor half GFN nor in CRUS) 65<=b<=105.

These bases are:

S83 (k=3), S85 (k=70), S88 (k=8), R67 (k=25), R75 (k=35), R91 (k=27), R94 (k=16), R100 (k=133).