mersenneforum.org - A Sierpinski/Riesel-like problem

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- sweety439 (https://www.mersenneforum.org/forumdisplay.php?f=137)

- - A Sierpinski/Riesel-like problem (https://www.mersenneforum.org/showthread.php?t=21839)

[QUOTE=sweety439;553809]A k-value which does not have covering set is proven composite by full algebraic factors if and only if [I]all[/I] n-values are algebraic (e.g. R4 k=1, R4 k=9, R9 k=1, R9 k=4, S8 k=27, S27 k=8)

A k-value which does not have covering set is proven composite by partial algebraic factors if and only if there is covering set for all n-values which is not algebraic (e.g. R12 k=25, R12 k=27, R19 k=4, R28 k=175, R30 k=1369, S55 k=2500)

Both cases of k-values are excluded from the conjectures.[/QUOTE]

In fact, there are two situations for which [I]all[/I] n-values are algebraic:

Case 1: (k and b are both perfect r-th power for an r>1 in Riesel side) (k and b are both perfect r-th power for an odd r>1 in Sierpinski side) (k is of the form 4*m^4 and b is perfect 4-th power in Sierpinski side), in this case the k-value is proven composite by full algebraic factors.

Case 2: k is rational power of b (in both sides), in this case the k-value is still included from the conjectures. (of course, in Riesel case if k and b are both perfect r-th power for an r>1, or in Sierpinski case if k and b are both perfect r-th power for an odd r>1, or in Sierpinski case k is of the form 4*m^4 and b is perfect 4-th power, then this k is excluded from the conjectures because of full algebra factors, no matter whether k is rational power of b or not)

* In Riesel side, if k is rational power of b, but k and b are not both perfect r-th power for all r>1, then this k is included from the conjectures (this case is generalized repunits to base b^(1/s))

* In Sierpinski side, if k is rational power of b, but k and b are not both perfect r-th power for all odd r>1, then there are three cases.... (let k = b^(r/s) with gcd(r,s) = 1)

** If the exponent of highest power of 2 dividing r < the exponent of highest power of 2 dividing s, then this k is included from the conjectures (this case is generalized repunits to negative base -b^(1/s))

** If the exponent of highest power of 2 dividing r >= the exponent of highest power of 2 dividing s, and the equation 2^x = r (mod s) has solutions, then this k is included from the conjectures (this case is GFN or half GFN to base b^(1/s), thus excluded from the weak conjectures)

** If the exponent of highest power of 2 dividing r >= the exponent of highest power of 2 dividing s, and the equation 2^x = r (mod s) has no solutions, then this k is excluded from the conjectures, since there are [I]no possible[/I] primes

[QUOTE=sweety439;553026]S13:

2 (2)
8 (4)
11 (564)
29 (10574)
281? (>5000)

S14:

2 (1)
6 (6)
22 (16)
29 (23)
61 (126)
73 (1182)
208 (>5000)

S15:

2 (1)
5 (2)
13 (10)
29 (30)
49 (112)
189 (190)
197 (464)
219 (1129)
341 (>5000)

S16:

2 (1)
3 (2)
5 (3)
18 (4)
23 (1074)
89 (>20000)

R13:

1 (5)
20 (10)
25 (15)
43 (77)
127 (95)
154 (469)
288 (109217)
337? (>5000)

R14:

1 (3)
2 (4)
5 (19698)
617? (>5000)

R15:

1 (3)
14 (14)
39 (16)
47 (>5000)

R16:

2 (1)
11 (2)
18 (3)
31 (12)
48 (15)
74 (638)
322 (4624)
443 (>1500000)[/QUOTE]

S17:

2 (47)
7 (190)
10 (1356)
53 (>4096)

S18:

2 (1)
3 (3)
13 (10)
37 (457)
122 (292318)
607? (>4096)

S19:

2 (1)
4 (3)
5 (78)
33 (286)
61 (>4096)

S20:

2 (1)
4 (2)
6 (15)
22 (106)
43 (2956)
277 (>4096)

S21:

2 (1)
3 (2)
12 (10)
67 (2490)
118 (19849)
139? (>4096)

S22:

2 (6)
10 (15)
23 (18)
70 (20)
77 (22)
128 (26)
137 (599)
173 (897)
346 (3180)
461 (16620)
740 (18137)
942 (18359)
1611 (738988)
1754? (>16800)

S23:

2 (1)
3 (3)
4 (342)
8 (119215)
61? (>4096)

S24:

2 (2)
5 (12)
12 (42)
61 (132)
202 (208)
224 (399)
319 (>4096)

S25: [k=5 is not half GFN]

2 (1)
5 (2)
12 (9)
40 (518)
61 (3104)
71 (>10000)

S26:

2 (1)
4 (2)
8 (35)
13 (68)
32 (318071)
65 (>1000000)

S27: [k=9 is half GFN]

2 (2)
7 (3)
21 (112)
33 (7876)
49? (>4096)

S28:

2 (1)
3 (7)
30 (10)
31 (17)
59 (282)
146 (47316)
871 (>1000000)

S29:

2 (1)
3 (2)
6 (4)
13 (6)
46 (24)
69 (35)
70 (348)
172 (468)
181 (778)
205 (>4096)

S30:

2 (1)
3 (3)
4 (6)
12 (1023)
242 (5064)
278 (>800000)

S31:

2 (2)
5 (1026)
43 (>6000)

S32:

3 (1)
5 (3)
7 (4)
9 (13)
26 (63)
47 (1223)
87 (1579)
94 (>1200000)

S64: [k=2 is not GFN]

2 (1)
6 (2)
11 (3222)
179 (>4096)

R17:

1 (3)
5 (60)
10 (117)
13 (1123)
29 (4904)
44 (6488)
103? (>4096)

R18:

1 (2)
9 (11)
32 (24)
41 (30)
50 (110)
78 (172)
151 (418)
324 (25665)
533? (>4096)

R19:

1 (19)
23 (108)
95 (872)
127 (>4096)

R20:

1 (3)
2 (10)
15 (21)
17 (22)
41 (28)
45 (154)
48 (162)
82 (>4096)

R21:

1 (3)
5 (4)
29 (98)
64 (2867)
606 (>4096)

R22:

1 (2)
8 (4)
16 (9)
25 (11)
29 (12)
55 (14)
70 (27)
89 (45)
91 (46)
106 (59)
185 (11433)
208 (>13000)

R23:

1 (5)
2 (6)
14 (52)
22 (55)
26 (214)
30 (1000)
107 (>4096)

R24:

1 (3)
14 (8)
19 (16)
53 (18)
69 (3896)
201 (>4096)

R25:

2 (2)
15 (4)
37 (17)
58 (26)
86 (1029)
181 (>4096)

R26:

1 (7)
23 (28)
25 (133)
32 (9812)
115 (520277)
178? (>4096)

R27:

2 (1)
3 (2)
9 (23)
23 (3742)
115 (>4096)

R28:

1 (2)
7 (26)
14 (47)
101 (53)
107 (74)
152 (75)
233 (>1000000)

R29:

1 (5)
2 (136)
52 (157)
122 (396)
151 (485)
152 (618)
269 (1352)
354 (>4096)

R30:

1 (2)
4 (3)
11 (30)
25 (34205)
225 (158755)
239 (337990)
659 (>500000)

R31:

1 (7)
3 (18)
5 (>6000)

R32:

2 (6)
3 (11)
13 (159)
29 (>2000000)

R64:

2 (1)
5 (2)
11 (9)
24 (3020)
157 (>4096)

[QUOTE=sweety439;553037]Sierpinski k=2:

3 (1)
12 (3)
17 (47)
38 (2729)
101 (192275)
218 (333925)
365? (>200000)

Sierpinski k=3:

2 (1)
5 (2)
18 (3)
28 (7)
43 (171)
79 (875)
83 (>8000)

Sierpinski k=4:

3 (1)
5 (2)
17 (6)
23 (342)
53 (>1610000)

Sierpinski k=5:

2 (1)
3 (2)
16 (3)
19 (78)
31 (1026)
137 (>2000)

Sierpinski k=6:

2 (1)
4 (2)
14 (6)
19 (14)
20 (15)
48 (27)
53 (143)
67 (4532)
108 (16317)
129 (16796)
212 (>400000)

Riesel k=1:

2 (2)
3 (3)
7 (5)
11 (17)
19 (19)
35 (313)
39 (349)
51 (4229)
91 (4421)
152 (270217)
185? (>66337)

Riesel k=2:

2 (1)
5 (4)
20 (10)
29 (136)
67 (768)
107 (21910)
170 (166428)
581 (>200000)

Riesel k=3:

2 (1)
3 (2)
23 (6)
31 (18)
42 (2523)
107 (4900)
295 (5270)
347 (>25000)

Riesel k=4:

2 (1)
7 (3)
23 (5)
43 (279)
47 (1555)
72 (1119849)
178? (>5000)

Riesel k=5:

2 (2)
8 (4)
14 (19698)
31? (>6000)

Riesel k=6:

2 (1)
13 (2)
21 (3)
48 (294)
119 (665)
154 (1989)
234 (>400000)[/QUOTE]

Riesel k=7:

2 (1)
3 (2)
7 (4)
28 (26)
31 (42)
41 (153)
68 (25395)

Riesel k=8:

2 (2)
7 (4)
29 (38)
68 (62)
71 (682)
97 (192335)
321 (>500000)

Riesel k=9:

2 (1)
11 (5)
18 (11)
27 (23)
38 (43)
71 (117)
88 (171)

Riesel:

[2,2293]

b=2, k<=2292

[3,1613]

b<=4, k<=1612

[5,1279]

b<=6, k<=1278

[7,679]

b<=7, k<=678

[8,239]

b<=10, k<=238

[11,201]

b<=14, k<=200

[15,47]

b<=30, k<=46

[31,5]

b<=177, k<=4

[178,4]

b<=184, k<=3

[185,1]

Riesel:

b=2: (see [URL="http://www.prothsearch.com/rieselprob.html"]http://www.prothsearch.com/rieselprob.html[/URL])

b=3:

k=97, n=3131 (k=291, n=3130, k=873, n=3129)
k=119, n=8972 (k=357, n=8971, k=1071, n=8970)
k=302, n=2091 (k=906, n=2090)
k=313, n=24761 (k=939, n=24760)
k=599, n=1240
k=811, n=1126
k=997, n=20847
k=1013, n=1233
k=1093, n=1297
k=1199, n=3876
k=1303, n=1384

b=4:

k=74, n=1276 (k=296, n=1275, k=1184, n=1274)
k=106, n=4553 (k=424, n=4552)
k=373, n=2508 (k=1492, n=2507)
k=659, n=400258
k=674, n=5838
k=751, n=6615
k=1103, n=2203
k=1159, n=5628
k=1171, n=2855
k=1189, n=3404
k=1211, n=12621
k=1524, n=1994

b=5:

k=86, n=2058 (k=430, n=2057)
k=428, n=9704
k=638, n=6974
k=662, n=14628
k=935, n=1560
k=1006, n=4197

b=6:

k=251, n=3008
k=1030, n=1199

b=7:

k=159, n=4896 (k=1113, n=4895)
k=197, n=181761
k=313, n=5907
k=367, n=15118
k=419, n=1052
k=429, n=3815
k=653, n=1051

b=8:

k=74, n=2632
k=151, n=2141
k=191, n=1198
k=203, n=1866
k=236, n=5258

b=9:

k=119, n=4486

b=11:

k=62, n=26202

b=14:

k=5, n=19698 (k=70, n=19697)

b=17:

k=13, n=1123
k=29, n=4904
k=44, n=6488

b=23:

k=30, n=1000

b=26:

k=32, n=9812

b=27:

k=23, n=3742

b=30:

k=25, n=34205

b=42:

k=3, n=2523

b=47:

k=4, n=1555

b=51:

k=1, n=4229

b=72:

k=4, n=1119849

b=91:

k=1, n=4421

b=107:

k=2, n=21910
k=3, n=4900

b=115:

k=4, n=4223

b=135:

k=1, n=1171

b=142:

k=1, n=1231

b=152:

k=1, n=270217

b=159:

k=3, n=2160

b=163:

k=4, n=2285

b=167:

k=4, n=1865

b=170:

k=2, n=166428

b=174:

k=1, n=3251

b=184:

k=1, n=16703

Riesel k=7:

2 (1)
3 (2)
7 (4)
28 (26)
31 (42)
41 (153)
68 (25395)
202? (>10000)

Riesel k=8:

2 (2)
7 (4)
29 (38)
68 (62)
71 (682)
97 (192335)
321 (>500000)

Riesel k=9:

2 (1)
11 (5)
18 (11)
27 (23)
38 (43)
71 (117)
88 (171)
107 (>10000)

Riesel k=10:

2 (1)
5 (3)
17 (117)
61 (1552)
80 (>400000)

Riesel k=11:

2 (2)
3 (22)
17 (46)
38 (766)
65 (>4096)

Riesel k=12:

2 (1)
3 (2)
8 (3)
10 (5)
18 (8)
31 (72)
43 (203)
65 (1193)
98 (3599)
153 (21659)
186 (112717)
263? (>100000)

[QUOTE=sweety439;553037]Sierpinski k=2:

3 (1)
12 (3)
17 (47)
38 (2729)
101 (192275)
218 (333925)
365? (>200000)

Sierpinski k=3:

2 (1)
5 (2)
18 (3)
28 (7)
43 (171)
79 (875)
83 (>8000)

Sierpinski k=4:

3 (1)
5 (2)
17 (6)
23 (342)
53 (>1610000)

Sierpinski k=5:

2 (1)
3 (2)
16 (3)
19 (78)
31 (1026)
137 (>2000)

Sierpinski k=6:

2 (1)
4 (2)
14 (6)
19 (14)
20 (15)
48 (27)
53 (143)
67 (4532)
108 (16317)
129 (16796)
212 (>400000)

Riesel k=1:

2 (2)
3 (3)
7 (5)
11 (17)
19 (19)
35 (313)
39 (349)
51 (4229)
91 (4421)
152 (270217)
185? (>66337)

Riesel k=2:

2 (1)
5 (4)
20 (10)
29 (136)
67 (768)
107 (21910)
170 (166428)
581 (>200000)

Riesel k=3:

2 (1)
3 (2)
23 (6)
31 (18)
42 (2523)
107 (4900)
295 (5270)
347 (>25000)

Riesel k=4:

2 (1)
7 (3)
23 (5)
43 (279)
47 (1555)
72 (1119849)
178? (>5000)

Riesel k=5:

2 (2)
8 (4)
14 (19698)
31? (>6000)

Riesel k=6:

2 (1)
13 (2)
21 (3)
48 (294)
119 (665)
154 (1989)
234 (>400000)[/QUOTE]

Sierpinski k=7: [base 7 is half GFN]

2 (2)
17 (190)
50 (516)
103 (>8000)

Sierpinski k=8:

3 (2)
6 (4)
23 (119215)
53 (227183)
86 (>1000000)

Sierpinski k=9: [base 3 and base 27 are half GFN]

2 (1)
7 (6)
31 (24)
43 (498)
63 (2162)
167 (>4096)

Sierpinski k=10:

2 (2)
11 (10)
17 (1356)
23 (3762)
173 (264234)
185 (>1000000)

Sierpinski k=11:

2 (1)
4 (2)
12 (3)
13 (564)
33 (593)
64 (3222)
68 (3947)
131 (>4096)

Sierpinski k=12:

2 (3)
7 (4)
21 (10)
24 (42)
30 (1023)
68 (656921)
163? (>400000)

[URL="https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures"]https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures[/URL] (2<=b<=128 or b=256, 512, 1024, 1<=k<1st CK (1<=k<=10000 for b=66, 120, 124; 1<=k<=30000 for b=126))

[URL="https://github.com/xayahrainie4793/first-4-Sierpinski-Riesel-conjectures"]https://github.com/xayahrainie4793/first-4-Sierpinski-Riesel-conjectures[/URL] (2<=b<=64 (b != 2, 3, 6, 15, 22, 24, 28, 30, 36, 40, 42, 46, 48, 52, 58, 60, 63) or b=100, 128, 256, 512, 1024, 1<=k<4th CK)

[URL="https://github.com/xayahrainie4793/all-k-1024"]https://github.com/xayahrainie4793/all-k-1024[/URL] (2<=b<=32 or b=64, 256, 1<=k<=1024)

Update zip file for 4<=b<=32 (b=2 and 3 are already in [URL="https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures"]https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures[/URL]) or b=64, 256, 1<=k<=1024, searched up to n=4096

Newest files:

[URL="https://docs.google.com/document/d/e/2PACX-1vQOXpXIq6Rxpt9re3HubJ4EGKft66bg2n63sCWkIWCy4tw9gBaU1UJn3aRv_zdLD9LMN4IXv01Lp3lY/pub"]Sierpinski problems[/URL]

[URL="https://docs.google.com/document/d/e/2PACX-1vQCSryWsLivzmUdW9_BKze8P7v1c8ZKlBzYIXNBgnTJJS4VRK2_5-s3-21wsdEjBxjBx6KE0G30YnOn/pub"]Riesel problems[/URL]