A Sierpinski/Riesel-like problem

[sweety439]

1,11
2,2
3,27
4,1
5,12
6,1
7,1
8,2
9,(partial algebra factors)
10,5
11,0
12,2
13,0
14,4
15,0
16,231

With CK=17

All k where k = m^2 and m = = 3 or 5 mod 8:
for even n let k = m^2 and let n = 2*q; factors to:
(m*137^q - 1) * (m*137^q + 1)
odd n:
factor of 2

This includes k = 9

k = 11, 13, 15 remain at n=2000

[sweety439]

1,2
138,1
275,1
412,2
549,4
686,1
823,1
960,1
1097,6
1234,28
1371,1
1508,2
1645,1
1782,2

With CK=1806

Only list k == 1 mod 137 since other k are already in CRUS

no remain k with k == 1 mod 137, totally list of the remain k is {408, 688, 831, 1074, 1743}, all remain at n=300K, see CRUS

[sweety439]

1,163
2,1
3,114
4,(partial algebra factors)
5,1

With CK=6

All k where k = m^2 and m = = 2 or 3 mod 5:
for even n let k = m^2 and let n = 2*q; factors to:
(m*139^q - 1) * (m*139^q + 1)
odd n:
factor of 5

This includes k = 4

Conjecture proven

[sweety439]

1,79
2,2
3,1
4,5
5,30
6,1
7,7
8,2
9,1
10,1
11,108
12,2
13,7
14,16
15,1
16,1
17,8
18,6
19,1
20,2
21,1
22,1
23,2
24,1
25,1
26,4
27,1
28,1
29,18
30,2
31,1
32,16
33,12
34,1
35,6
36,1
37,1
38,448
39,2
40,9
41,8
42,1
43,3
44,2
45,1

With CK=46

Conjecture proven

[sweety439]

A (Sierpinski/Riesel) base b has 2-cover if and only if b+1 is not prime or prime power

[sweety439]

The formula for Sierpinski conjectures in CRUS is k*b^n+1
The formula for Riesel conjectures in CRUS is k*b^n-1
The formula for Sierpinski conjectures in this project is (k*b^n+1)/gcd(k+1,b-1)
The formula for Riesel conjectures in this project is (k*b^n-1)/gcd(k-1,b-1)

[sweety439]

The reason for (k=27 is excluded from S8 in this project) (k=4 is excluded from S16 in this project) (k=1 is excluded from R4 in this project) (k=1 is excluded from R8 in this project) (k=1 is excluded from R16 in this project) (k=1 is excluded from R36 in this project) (k=1 is excluded from R100 in this project) (k=1 is excluded from R128 in this project) is the same as (k=1 is excluded from R14 in CRUS) (k=1 is excluded from R18 in CRUS) (k=1 is excluded from R20 in CRUS) (k=1 is excluded from R24 in CRUS) (k=1 is excluded from R30 in CRUS) (k=1 is excluded from R32 in CRUS) (k=1 is excluded from R38 in CRUS) (k=1 is excluded from R42 in CRUS)

[sweety439]

searched to n=2000, see the text file for the status, 0 if no (probable) prime found for this k

CK=285

Only k = 201 remain at n=2000

[sweety439]

1,1231
2,1
3,26
4,3
5,1
6,3
7,1
8,7
9,1
10,2
11,14

With CK=12

Conjecture proven

[sweety439]

1,3
2,2
3,16
4,1

With CK=5

Conjecture proven

[sweety439]

1,(full algebra factors)
2,24
3,1
4,(full algebra factors)
5,1
6,1
7,5
8,1
9,(full algebra factors)
10,1
11,1
12,1
13,1
14,4
15,10
16,(full algebra factors)
17,1
18,1
19,4
20,1
21,1
22,1
23,134
24,2
25,(full algebra factors)
26,5
27,2
28,2
29,4
30,519
31,1
32,3
33,1
34,8
35,1
36,(full algebra factors)
37,3
38,1
39,964
40,1
41,1
42,1
43,2
44,6
45,3
46,97
47,2
48,1
49,(full algebra factors)
50,2
51,3
52,1
53,1
54,8
55,1
56,1
57,20
58,35

With CK=59

All k = m^2 for all n; factors to:
(m*12^n - 1) * (m*12^n + 1)

This includes k = 1, 4, 9, 16, 25, 36, 49

Conjecture proven