A Sierpinski/Riesel-like problem

[sweety439]

Originally Posted by sweety439

Found these (probable) primes:

(44*1024^1933+1)/3
(13*1024^1167-1)/3
(43*1024^2290-1)/3

These k's are still remain:

S1024: 4, 16, 29, 38, 56
R1024: 29, 31, 56, 61

Note that both sides have k=29 and k=56 remain, but the divisors (i.e. gcd(k+-1,b-1)) for two sides for these k's aren't the same.

No additional primes found for SR1024, these k's are likely tested to n=3K.

[sweety439]

1M < (CK for S66) <= 21314443
1M < (CK for S120) <= 374876369
1M < (CK for R66) <= 101954772
1M < (CK for R120) <= 166616308

[sweety439]

Originally Posted by sweety439

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.

If n is even, let n=2*q, then (400*88^n-1)/3 can be factored to:

(20*88^q-1) * ((20*88^q+1)/3)

Since n>=1, thus we have q>=1, and

20*88^q-1 >= 20*88^1-1 = 1759 > 1
(20*88^q+1)/3 >= (20*88^1+1)/3 = 587 > 1

Thus, this factorization is not trivial, and hence (400*88^n-1)/3 is composite for all even n. (Xayah)

If n is odd, then n is congruent to either 1, 3, or 5 mod 6, however:

If n = 1 mod 6, then (400*88^n-1)/3 is divisible by 3, and since n>=1, we have (400*88^n-1)/3 >= (400*88^1-1)/3 = 11733 > 3, thus (400*88^n-1)/3 is composite for all n = 1 mod 6.

If n = 3 mod 6, then (400*88^n-1)/3 is divisible by 7 and greater than 7, thus (400*88^n-1)/3 is composite for all n = 3 mod 6.

If n = 5 mod 6, then (400*88^n-1)/3 is divisible by 13 and greater than 13, thus (400*88^n-1)/3 is composite for all n = 5 mod 6.

Thus, (400*88^n-1)/3 is composite for all odd n. (Rakan)

By (Xayah) and (Rakan), (400*88^n-1)/3 is composite for all n>=1.

[sweety439]

Originally Posted by sweety439

The records for the n are: (excluding the k's with covering set and the k's proven composite by full/partial algebra factors)

S2:

1 (1)
4 (2)
12 (3)
16 (4)
19 (6)
31 (8)
47 (583)
383 (6393)
2897 (9715)
3061 (33288)
4847 (3321063)
5359 (5054502)
10223 (31172165)
21181 (>31600000)

S3:

1 (1)
5 (2)
16 (3)
17 (6)
21 (8)
41 (4892)
621 (>10000)

S4:

1 (1)
6 (2)
19 (3)
30 (4)
51 (46)
86 (108)
89 (167)
94 (291)
186 (10458)

(next record k is > 794)

S5:

1 (1)
3 (2)
18 (3)
19 (4)
34 (8)
40 (1036)
61 (6208)
181 (>12000)

S6:

1 (1)
8 (4)
20 (5)
53 (7)
67 (8)
97 (9)
117 (23)
136 (24)
160 (3143)
1296 (>=268435454)

S7:

1 (4)
9 (6)
21 (124)
101 (216)
121 (252)
141 (1044)

(next record k is > 209)

S8:

2 (1)
3 (2)
13 (4)
31 (20)
68 (115)
94 (194)
118 (820)
173 (7771)
256 (>=2863311528)

S9:

1 (1)
6 (2)
17 (3)
21 (4)
26 (6)
40 (9)
41 (2446)
311? (>2000)

S10:

1 (1)
8 (2)
9 (3)
22 (6)
34 (26)
100 (>=33554430)

S11:

1 (2)
10 (10)
20 (35)
45 (40)
47 (545)
194 (>2000)

S12:

1 (1)
2 (3)
12 (>=33554431)

R2:

1 (2)
13 (3)
14 (4)
43 (7)
44 (24)
74 (2552)
659 (800516)
2293 (>8300000)

R3:

1 (3)
11 (22)
71 (46)
97 (3131)
119 (8972)
313 (24761)
1613 (>50000)

R4:

2 (1)
7 (2)
39 (12)
74 (1276)
106 (4553)
659 (400258)

(next record k is > 1114)

R5:

1 (3)
2 (4)
31 (5)
32 (8)
34 (163)
86 (2058)
428 (9704)
638? (>2000)

R6:

1 (2)
37 (4)
54 (6)
69 (10)
92 (49)
251 (3008)
1597 (>5000000)

R7:

1 (5)
31 (18)
59 (32)
73 (127)
79 (424)
139 (468)
159 (4896)
197 (181761)

(next record k is > 457)

R8:

2 (2)
5 (4)
11 (18)
37 (851)
74 (2632)
236 (5258)
239? (>2000)

R9:

2 (1)
11 (11)
53 (536)
119 (4486)
386 (>5000)

R10:

1 (2)
12 (5)
32 (28)
89 (33)
98 (90)
109 (136)
121 (483)
406 (772)
450 (11958)
505 (18470)
1231 (37398)
1803 (45882)
1935 (51836)
2452 (>554789)

R11:

1 (17)
32 (18)
39 (22)
62 (26202)
201? (>2000)

R12:

1 (2)
23 (3)
24 (4)
46 (194)
157 (285)
298 (1676)
1037 (>5000)

If we exclude GFN's and half GFN's, then the Sierpinski bases should be changed as follow:

S2:

3 (1)
7 (2)
12 (3)
19 (6)
31 (8)
47 (583)
383 (6393)
2897 (9715)
3061 (33288)
4847 (3321063)
5359 (5054502)
10223 (31172165)
21181 (>31600000)

S3:

2 (1)
5 (2)
16 (3)
17 (6)
21 (8)
41 (4892)
621 (>10000)

S4:

2 (1)
6 (2)
19 (3)
30 (4)
51 (46)
86 (108)
89 (167)
94 (291)
186 (10458)

(next record k is > 794)

S5:

2 (1)
3 (2)
18 (3)
19 (4)
34 (8)
40 (1036)
61 (6208)
181 (>12000)

S6:

2 (1)
8 (4)
20 (5)
53 (7)
67 (8)
97 (9)
117 (23)
136 (24)
160 (3143)
1814 (>12000)

S7:

2 (1)
5 (2)
9 (6)
21 (124)
101 (216)
121 (252)
141 (1044)

(next record k is > 209)

S8:

3 (2)
13 (4)
31 (20)
68 (115)
94 (194)
118 (820)
173 (7771)
259 (27626)
370? (>10000)

S9:

2 (1)
6 (2)
17 (3)
21 (4)
26 (6)
40 (9)
41 (2446)
311? (>2000)

S10:

2 (1)
8 (2)
9 (3)
22 (6)
34 (26)
269 (>24800)

S11:

2 (1)
4 (2)
10 (10)
20 (35)
45 (40)
47 (545)
194 (>2000)

S12:

2 (3)
17 (78)
30 (144)
37 (199)
261 (644)
378 (2388)
404 (714558)
563? (>5000)

[sweety439]

Originally Posted by sweety439

Found the all CK<=10000 for all extended Sierpinski/Riesel problem bases 129<=b<=1024, the text file lists "NA" if and only if CK>10000 for this Sierpinski/Riesel base.

Note: I only tested the primes p<=30000, and I only searched (k*b^n+-1)/gcd(k+-1,b-1) for exponent n<=1500.

Found more CK:

S136: CK=29180
S150: CK=49074
S192: CK=7879
S196: CK=16457
R136: CK=22195
R150: CK=49074
R192: CK=13897
R196: CK=1267

I still cannot find the CK for SR66 and SR120 :(

[sweety439]

Originally Posted by sweety439

Found more CK:

S136: CK=29180
S150: CK=49074
S192: CK=7879
S196: CK=16457
R136: CK=22195
R150: CK=49074
R192: CK=13897
R196: CK=1267

I still cannot find the CK for SR66 and SR120 :(

Also,

S228: CK=1146
S232: CK=2564
S238: CK=34571
R228: CK=16718
R232: CK=27760
R238: CK=17926

Now, all CK for Sierpinski/Riesel bases b<=256 with b = 1, 3, 4, 5 (mod 6) were found, except SR190, I am now looking for the CK for SR190.

[sweety439]

Originally Posted by sweety439

Also,

S228: CK=1146
S232: CK=2564
S238: CK=34571
R228: CK=16718
R232: CK=27760
R238: CK=17926

Now, all CK for Sierpinski/Riesel bases b<=256 with b = 1, 3, 4, 5 (mod 6) were found, except SR190, I am now looking for the CK for SR190.

Now, all CK for bases b<=256 are found except (in both two sides) b = 66, 120, 156, 180, 190, 210, 222, 240.

[sweety439]

Originally Posted by sweety439

Also,

S228: CK=1146
S232: CK=2564
S238: CK=34571
R228: CK=16718
R232: CK=27760
R238: CK=17926

Now, all CK for Sierpinski/Riesel bases b<=256 with b = 1, 3, 4, 5 (mod 6) were found, except SR190, I am now looking for the CK for SR190.

Also found:

S190: CK=2157728
S222: CK=333163
R190: CK=626861
R222: CK=88530

I have not tested other bases (b = 156, 180, 210, 240).

[sweety439]

My computer is now searching for generalized repunit primes (b^n-1)/(b-1) (see http://mersenneforum.org/showthread.php?t=21808), I will reserve this project (extended SR problems) after these searched were done.

[sweety439]

In Riesel conjectures, if k=m^2 or k*b=m^2 and m and b satisfy at least one of these conditions, then this k should be excluded from the Riesel base b problem, since it has algebraic factors for even n (for the k=m^2 case) or odd n (for k*b=m^2 case) and it has a single prime factor for odd n (for the k=m^2 case) or even n (for the k*b=m^2 case).

m                     b
= 2 or 3 mod 5        = 4 mod 5
= 5 or 8 mod 13       = 12 mod 13
= 3 or 5 mod 8        = 9 mod 16
= 4 or 13 mod 17      = 16 mod 17
= 12 or 17 mod 29     = 28 mod 29
= 7 or 9 mod 16       = 17 mod 32
= 6 or 31 mod 37      = 36 mod 37
= 9 or 32 mod 41      = 40 mod 41
= 23 or 30 mod 53     = 52 mod 53
= 11 or 50 mod 61     = 60 mod 61
= 15 or 17 mod 32     = 33 mod 64
= 27 or 46 mod 73     = 72 mod 73
= 34 or 55 mod 89     = 88 mod 89
= 22 or 75 mod 97     = 96 mod 97
= 10 or 91 mod 101    = 100 mod 101
= 33 or 76 mod 109    = 108 mod 109
= 15 or 98 mod 113    = 112 mod 113
= 31 or 33 mod 64     = 65 mod 128
= 37 or 100 mod 137   = 136 mod 137
= 44 or 105 mod 149   = 148 mod 149
= 28 or 129 mod 157   = 156 mod 157
= 80 or 93 mod 173    = 172 mod 173
= 19 or 162 mod 181   = 180 mod 181
= 81 or 112 mod 193   = 192 mod 193
= 14 or 183 mod 197   = 196 mod 197
= 107 or 122 mod 229  = 228 mod 229
= 89 or 144 mod 233   = 232 mod 233
= 64 or 177 mod 241   = 240 mod 241
= 63 or 65 mod 128    = 129 mod 256
= 16 or 241 mod 257   = 256 mod 257
etc.

Since these k's proven composite by partial algebraic factors.

[sweety439]

Originally Posted by sweety439

In Riesel conjectures, if k=m^2 and m and b satisfy at least one of these conditions, then this k should be excluded from the Riesel base b problem.

m                  b
= 2 or 3 mod 5     = 4 mod 5
= 5 or 8 mod 13    = 12 mod 13
= 3 or 5 mod 8     = 9 mod 16
= 4 or 13 mod 17   = 16 mod 17
= 12 or 17 mod 29  = 28 mod 29
= 7 or 9 mod 16     = 17 mod 32
= 6 or 31 mod 37   = 36 mod 37
= 9 or 32 mod 41   = 40 mod 41
= 23 or 30 mod 53  = 52 mod 53
= 11 or 50 mod 61  = 60 mod 61
= 15 or 17 mod 32  = 33 mod 64
etc.

Of course, if there is an r>1 such that both k and b are perfect r-th powers, then this k should be excluded from the Riesel base b problem. Besides, if there is an odd r>1 such that both k and b are perfect r-th powers, then this k should be excluded from the Sierpinski base b problem. Besides, if k is of the form 4*m^4 and b is a perfect 4th power, then this k should be excluded from the Sierpinski base b problem.

Since these k's proven composite by full algebraic factors.