Sierpinski and Riesel number (Fixed k, Variable base)

[robert44444uk]

Thinking a bit about this:

The simplest covering set comprises 2 primes with order base b of 2. One provides cover for n 1,3,5,7... and the other for 2,4,6,8...

It is simple to prove that there are no two such primes.

Covering n=1 implies that (2.i^1+1)modp == 0modp and (2.i^3+1)modp==0modp

or 0modp == (2i^3-2i)modp
0modp == 2.i.(i-1).(i+1)modp
therefore i=1 and (2.1^2+1)modp==3mod(p)==0mod(p) which is only solvable for p=3, which is only 1 prime, and in position 1,3,5... this is base=1mod3 and position 2,4,6... is 2mod3

A second covering set can be described using 3 and two other primes q and r which cover either position 1 and 5 or 2 and 6, with 3 covering 2,4,6.. or 1,3,5.. . Such q,r are of the form 4j+1, where j is an integer.

Let us look at 3 covering positions 2,4,6.. and q covering 1,5,9...

Then (2.b^1+1)modq==(2.b^5+1)modq is solvable for 5, in position 1 as 2mod5 and position 2 as 3mod5

Trying to find prime r of the form 4j+1 requires a solution to the equation

A 4th root of unity modr equals either (r-1)/2 or the solution to (2.x^3+1)modr == 0modr

I checked the first few 4j+1 primes and found no solutions except 5:

1. r
2. a 4th root of unity modr
3. another 4th root of unity modr
4. (r-1)/2
5. solution to (2.x^3+1)modr==0modr
**= no solution

5 2 3 2 3
13 5 8 6 **
17 4 13 8 2
29 12 17 14 10
37 6 31 18 **
41 9 32 20 8
53 23 30 26 50
61 11 50 30 **
73 27 46 36 **
89 34 55 44 50
97 22 75 48 **
101 10 91 50 66
109 33 76 54 62
113 15 98 56 53
137 37 100 68 130
149 44 105 74 83
157 28 129 78 15
173 80 93 86 101
181 19 162 90 **
193 81 112 96 **
197 14 183 98 75
229 107 122 114 7
233 89 144 116 195
241 64 177 120 **
257 16 241 128 225
269 82 187 134 19

Makes me think that there may be no r that can satisfy the criteria, and that any covering set (if one exists) has to be more complex.

Perhaps someone can find an r or prove that I am comparing apples and oranges.

[robert44444uk]

Originally Posted by Citrix

I don't think that 2 can be a sierpinki/riesel number for any base. Nor can any of the low k values.

4 is the lowest k I have found, which is sierpinski and riesel quite frequently for bases where b^2-1 produces two primitive prime factors.

For example 4*14^n+/-1 never prime. Primitive primes factors of 14^2-1 are 3 and 5.

Maybe k=1 can be sierpinski/ riesel. May have a look at that.

[sweety439]

For fixed k, find the smallest base b such that all numbers of the form k*b^n+1 (k*b^n-1) are composite.

If k is of the form 2^n-1 (2^n+1), except k=1 (k=9), it is conjectured for every nontrivial base b, there is a prime of the form k*b^n+1 (k*b^n-1).

However, for all other k's, there is a base b such that all numbers of the form k*b^n+1 (k*b^n-1) are composite.

S = conjectured smallest base b such that k is a Sierpinski number.

k S remaining bases b with no known primes
1 none {38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016, 1026, ...} (b=m^r with odd r>1 proven composite by full algebraic factors)
2 201446503145165177 (?) {218, 236, 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004, ...}
3 none {718, 912, ...}
4 14 proven
5 140324348 {308, 326, 512, 824, ...}
6 34 proven
7 none {1004, ...}
8 20 proven (b=8 proven composite by full algebraic factors)
9 177744 {592, 724, 884, ...}
10 32 proven
11 14 proven
12 142 {12}
13 20 proven
14 38 proven
15 none {398, 650, 734, 874, 876, 1014, ...}
16 38 {32}
17 278 {68, 218}
18 322 {18, 74, 227, 239, 293}
19 14 proven
20 56 proven
21 54 proven
22 68 {22}
23 32 proven
24 114 {79}
25 38 proven
26 14 proven
27 90 {62}
28 86 {41}
29 20 proven
30 898
31 none
32 92 {87} (b=32 proven composite by full algebraic factors)

R = conjectured smallest base b such that k is a Riesel number.

k R remaining bases b with no known primes
1 none proven
2 none {303, 522, 578, 581, 992, 1019, ...}
3 none {588, 972, ...}
4 14 proven (b=9 proven composite by full algebraic factors)
5 none {338, 998, ...}
6 34 proven (b=24 proven composite by partial algebraic factors)
7 9162668342 {308, 392, 398, 518, 548, 638, 662, 848, 878, ...}
8 20 proven
9 none {378, 438, 536, 566, 570, 592, 636, 688, 718, 808, 830, 852, 926, 990, 1010, ...} (b=m^2 proven composite by full algebraic factors, b=4 mod 5 proven composite by partial algebraic factors)
10 32 proven
11 14 proven
12 142 proven
13 20 proven
14 8 proven
15 8241218 {454, 552, 734, 856, ...}
16 50 proven (b=9 proven composite by full algebraic factors, b=33 proven composite by partial algebraic factors)
17 none {98, 556, 650, 662, 734, ...}
18 203 {174} (b=50 proven composite by partial algebraic factors)
19 14 proven
20 56 proven
21 54 proven
22 68 {38, 62}
23 32 proven
24 114 proven
25 38 proven (b=36 proven composite by full algebraic factors, b=12 proven composite by partial algebraic factors)
26 14 proven
27 90 {34} (b=8 and 64 proven composite by full algebraic factors, b=12 proven composite by partial algebraic factors)
28 86 {74}
29 20 proven
30 898
31 362 {80, 84, 122, 278, 350}
32 92 {54, 71, 77}

[sweety439]

S = conjectured smallest base b such that k is a Sierpinski number.

k S cover set
1 none
2 201446503145165177 (?) {3, 5, 17, 257, 641, 65537, 6700417} period=64
3 none
4 14 {3, 5} period=2
5 140324348 {3, 13, 17, 313, 11489} period=16
6 34 {5, 7} period=2
7 none
8 20 {3, 7} period=2
9 177744 {5, 17, 41, 193} period=8
10 32 {3, 11} period=2
11 14 {3, 5} period=2
12 142 {11, 13} period=2
13 20 {3, 7} period=2
14 38 {3, 13} period=2
15 none
16 38 {3, 5, 17} period=4
17 278 {3, 5, 29} period=4
18 322 {17, 19} period=2
19 14 {3, 5} period=2
20 56 {3, 19} period=2
21 54 {5, 11} period=2
22 68 {3, 23} period=2
23 32 {3, 11} period=2
24 114 {5, 23} period=2
25 38 {3, 13} period=2
26 14 {3, 5} period=2
27 90 {7, 13} period=2
28 86 {3, 29} period=2
29 20 {3, 7} period=2
30 898 {29, 31} period=2
31 none
32 92 {3, 31} period=2
33 5592 {5, 17, 109} period=4
34 14 {3, 5} period=2
35 50 {3, 17} period=2
36 68 {5, 7, 13, 31, 37} period=12
37 56 {3, 19} period=2
38 98 {3, 5, 17} period=4
39 94 {5, 19} period=2
40 122 {3, 41} period=2
41 14 {3, 5} period=2
42 1762 {41, 43} period=2
43 32 {3, 11} period=2
44 128 {3, 43} period=2
45 252 {11, 23} period=2
46 140 {3, 47} period=2
47 8 {3, 5, 13} period=4
48 328 {7, 47} period=2
49 14 {3, 5} period=2
50 20 {3, 7} period=2
51 64 {5, 13} period=2
52 158 {3, 53} period=2
53 38 {3, 13} period=2
54 264 {5, 53} period=2
55 20 {3, 7} period=2
56 14 {3, 5} period=2
57 202 {7, 29} period=2
58 176 {3, 59} period=2
59 144 {5, 29} period=2
60 3598 {59, 61} period=2
61 92 {3, 31} period=2
62 182 {3, 61} period=2
63 none
64 29 {3, 5} period=2

R = conjectured smallest base b such that k is a Riesel number.

k R cover set
1 none
2 none
3 none
4 14 {3, 5} period=2
5 none
6 34 {5, 7} period=2
7 9162668342 {3, 5, 17, 1201, 169553} period=16
8 20 {3, 7} period=2
9 none
10 32 {3, 11} period=2
11 14 {3, 5} period=2
12 142 {11, 13} period=2
13 20 {3, 7} period=2
14 8 {3, 5, 13} period=4
15 8241218 {7, 17, 113, 1489} period=8
16 50 {3, 17} period=2
17 none
18 203 {5, 13, 17} period=4
19 14 {3, 5} period=2
20 56 {3, 19} period=2
21 54 {5, 11} period=2
22 68 {3, 23} period=2
23 32 {3, 11} period=2
24 114 {5, 23} period=2
25 38 {3, 13} period=2
26 14 {3, 5} period=2
27 90 {7, 13} period=2
28 86 {3, 29} period=2
29 20 {3, 7} period=2
30 898 {29, 31} period=2
31 362 {3, 7, 13, 37, 331} period=12
32 92 {3, 31} period=2
33 none
34 14 {3, 5} period=2
35 50 {3, 17} period=2
36 184 {5, 37} period=2
37 56 {3, 19} period=2
38 110 {3, 37} period=2
39 94 {5, 19} period=2
40 122 {3, 41} period=2
41 14 {3, 5} period=2
42 1762 {41, 43} period=2
43 32 {3, 11} period=2
44 128 {3, 43} period=2
45 252 {11, 23} period=2
46 140 {3, 47} period=2
47 68 {3, 23} period=2
48 328 {7, 47} period=2
49 14 {3, 5} period=2
50 20 {3, 7} period=2
51 64 {5, 13} period=2
52 158 {3, 53} period=2
53 38 {3, 13} period=2
54 264 {5, 53} period=2
55 20 {3, 7} period=2
56 14 {3, 5} period=2
57 202 {7, 29} period=2
58 176 {3, 59} period=2
59 86 {3, 29} period=2
60 3598 {59, 61} period=2
61 68 {3, 7, 13, 31} period=6
62 182 {3, 61} period=2
63 36858 {5, 31, 397} period=4
64 14 {3, 5} period=2

[sweety439]

See http://mersenneforum.org/showthread.php?t=21951 for more information.

[sweety439]

Originally Posted by Citrix

Has there been any work done to find S/R for fixed k and variable base?

eg. What is the lowest base=b for k=2 such that 2*b^n+1 is never prime?

What k's can never be sierpinski numbers? Is there any proof to this.

How does one generate the sierpinski base for a given k?

(The same questions for Riesel side)

This base is probably 201446503145165177.