A Sierpinski/Riesel-like problem

[sweety439]

In this case, although (k*b^n+1)/gcd(k+1,b-1) has neither covering set nor algebra factors, but this form still cannot have a prime, thus this case is also excluded in the conjectures. (this situation only exists in the Sierpinski side)

b = q^m, k = q^r, where q is not of the form t^s with odd s>1, and m and r have no common odd prime factor, and the exponent of highest power of 2 dividing r >= the exponent of highest power of 2 dividing m, and the equation 2^x = r (mod m) has no solution.

Examples:

b = q^7, k = q^r, where r = 3, 5, 6 (mod 7).
b = q^14, k = q^r, where r = 6, 10, 12 (mod 14).
b = q^15, k = q^r, where r = 7, 11, 13, 14 (mod 15).
b = q^17, k = q^r, where r = 3, 5, 6, 7, 10, 11, 12, 14 (mod 17).
b = q^21, k = q^r, where r = 5, 10, 13, 17, 19, 20 (mod 21)
b = q^23, k = q^r, where r = 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22 (mod 23)
b = q^28, k = q^r, where r = 12, 20, 24 (mod 28)
b = q^30, k = q^r, where r = 14, 22, 26, 28 (mod 30)
b = q^31, k = q^r, where r = 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 (mod 31)
b = q^33, k = q^r, where r = 5, 7, 10, 13, 14, 19, 20, 23, 26, 28 (mod 33)
etc.

(these are all examples for m<=33)

[sweety439]

In the Sierpinski side, if k is a rational power of b, then this number is a generalized Fermat number in base q (q is the largest number such that both b and k are integer powers of q) (if gcd(k+1,b-1) = 1) or a half generalized Fermat number in base q (q is the largest number such that both b and k are integer powers of q) (if gcd(k+1,b-1) = 2) or a generalized repunit in base -q (q is the largest number such that both b and k are integer powers of q) (if gcd(k+1,b-1) is neither 1 nor 2). Besides, in the Riesel side, if k is a rational power of b, then this number is a generalized repunit in base q (q is the largest number such that both b and k are integer powers of q), if b is not a perfect power, then if k is a rational power of b, then k is always a integer power of b, thus this number is a base b repunit (all such k's are in the same family), if b is a perfect power and this number has no algebra factors, then this number is a base q repunit, where q is the largest number such that both b and k are integer powers of q.

In the Riesel side (we only consider such k's < b, since all k's > b is a multiple of b, which are in the same family with smaller k), the correspond primes for non-perfect power b (in the case, k must be 1) are already listed in http://oeis.org/A084740, and the correspond primes for perfect powers b<=144 are: (exclude the forms with algebra factors, i.e. if b = q^m and k = q^r, then gcd(m,r) = 1) (some cases are k's > CK for the base)

b    k    prime
4    2    (2*4^1-1)/1
8    2    (2*8^2-1)/1
8    4    (4*8^1-1)/1
9    3    (3*9^1-1)/2
16   2    (2*16^1-1)/1
16   8    (8*16^1-1)/1
25   5    (5*25^1-1)/4
27   3    (3*27^2-1)/2
27   9    (9*27^23-1)/2
32   2    (2*32^6-1)/1
32   4    (4*32^1-1)/1
32   8    (8*32^2-1)/1
32   16   (16*32^3-1)/1
36   6    (6*36^1-1)/5
49   7    (7*49^2-1)/6
64   2    (2*64^1-1)/1
64   32   (32*64^2-1)/1
81   3    (3*81^3-1)/2
81   27   (27*81^1-1)/2
100  10   (10*100^9-1)/9
121  11   (11*121^8-1)/10
125  5    (5*125^2-1)/4
125  25   (25*125^3-1)/4
128  2    (2*128^18-1)/1
128  4    (4*128^15-1)/1
128  8    (8*128^2-1)/1
128  16   (16*128^459-1)/1
128  32   (32*128^2-1)/1
128  64   (64*128^1-1)/1
144  12   (12*144^1-1)/11

[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

[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

[sweety439]

Some extended Sierpinski/Riesel bases are "proven", but with some probable primes that have not been certified, these bases <=64 and probable primes are shown below:

base     k       probable prime
S61      43      (43*61^2788+1)/4
S61      62      (62*61^3698+1)/3
S64      11      (11*64^3222+1)/3
R7       159     (159*7^4896-1)/2
R7       197     (197*7^181761-1)/2
R7       313     (313*7^5907-1)/6
R7       367     (367*7^15118-1)/6
R17      29      (29*17^4904-1)/4
R51      1       (1*51^4229-1)/50

Update the word file to show which primes are only probable primes that have not been certified.

[sweety439]

These files are the status of S5, S9 and S11 for all k<=1024. (tested up to n=2000) (0 if there are no (probable) primes for n<=2000)

These k's with covering set, thus excluded in the files:

S5: k = 7, 11 (mod 24), covering set {2, 3}
S9: k = 31, 39 (mod 80), covering set {2, 5}
S11: k = 5, 7 (mod 12), covering set {2, 3}

[sweety439]

These files are the status of R5, R9 and R11 for all k<=1024. (tested up to n=2000) (0 if there are no (probable) primes for n<=2000)

These k's with covering set, thus excluded in the files:

R5: k = 13, 17 (mod 24), covering set {2, 3}
R9: k = 41, 49 (mod 80), covering set {2, 5}, also square k's with full algebra factors
R11: k = 5, 7 (mod 12), covering set {2, 3}

[sweety439]

Originally Posted by kar_bon

To give you a number of the work that stands for finding (197*7^181761-1)/2 as PRP:

all timings of those ~1500 checked candidates for PRP with pfgw on a i7-2600 3,4 GHz 64Bit stystem doing in one core took me ~158 hours and those candidates eliminated by trial factoring are not included here.

I got my own searches and factoring and this was only of some interest to find some high PRPs.

Your researches on your own RS-conj. are at n=1K or 6K only by now, so do some work say n=50K for all bases. Primes searching and especially those projects need patience and the results will come.

The problem on your project is (as Gary mentioned), you have to determine the GCD for every k-value of any base to search for. It would by better to give those GCDs in the tables: sorting k-values by GCD.

I cannot run those programs.

Can you reserve S10 k=269 and S25 k=71?

[sweety439]

Originally Posted by sweety439

Of course, we can also find the primes for the k's > CK. e.g. we can try to prove the 2nd conjecture, 3rd conjecture, 4th conjecture, ..., for a fixed Sierpinski/Riesel base.

For example, the 2nd conjectured k for S2 is 271129, and this conjecture is being worked on https://www.primegrid.com/forum_thread.php?id=1750. Besides, the 2nd conjectured k for R4 is 919, and this conjecture is proven, but with one non-certified probable prime (751*4^6615-1)/3. (for the (probable) prime for the 2nd conjecture for R4, see post #81)

However, in this project, we only decide to prove the "1st conjecture". Thus, in this project, we only consider the k's < CK.

You can also try to prove the first, second, third, ... conjectured k for one specific base with a lot of small conjectured k (S5 for example has conjectured k at k = 7, 11, 31, 35, 55, 59, 79, 83, ..., R5 for example has conjectured k at k = 13, 17, 37, 41, 61, 65, 85, 89, ...)

Sierpinski and Riesel bases 5, 8, 9, 11, 13, 14, 16 would be interesting bases to attack to prove the 2nd/3rd/etc. conjectured k's since their 1st one is so low and was already easily proven.

Also, the 2nd/3rd/etc. conjecture for R10 has been worked in http://www.worldofnumbers.com/Append...s%20to%20n.txt (k mod 9 = 1) and https://www.rose-hulman.edu/~rickert...siteseq/#b10d3 (k mod 9 = 4 or 7), R10 has conjectured k at k = 334, 1585, 1882, 3340, 3664, 7327, 8425, 9208, 10176, ... (k = 343 and 3430 proven composite by partial algebraic factors), and this base has 3 k's remain for k < 10176: 2452, 4421 and 5428. (test limit: k = 2452 at n=554K, k = 4421 at n=1.76M, k = 5428 at n=300K)

[sweety439]

Update the primes for SR22, R28, SR46, SR58 and SR63 for the k's such that gcd(k+-1,b-1) = 1 (+ for Sierpinski, - for Riesel), compare with CRUS.

See http://www.mersennewiki.org/index.ph..._definition%29 (Sierpinski) and http://www.mersennewiki.org/index.ph..._definition%29 (Riesel).

[sweety439]

Reserve SR42, SR48 and SR60 (only for the k's not in CRUS, i.e. gcd(k+-1,b-1) > 1),