mersenneforum.org A Sierpinski/Riesel-like problem
 Register FAQ Search Today's Posts Mark Forums Read

2020-08-15, 19:15   #947
sweety439

Nov 2016

45018 Posts

Quote:
 Originally Posted by sweety439 A k-value which does not have covering set is proven composite by full algebraic factors if and only if all 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.
In fact, there are two situations for which all 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 no possible primes

Last fiddled with by sweety439 on 2020-08-20 at 04:38

2020-08-15, 19:44   #948
sweety439

Nov 2016

236910 Posts

Quote:
 Originally Posted by sweety439 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)
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)

 2020-08-15, 21:54 #949 sweety439     Nov 2016 23×103 Posts 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)
2020-08-15, 22:04   #950
sweety439

Nov 2016

23×103 Posts

Quote:
 Originally Posted by sweety439 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)
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)

Last fiddled with by sweety439 on 2020-08-15 at 22:05

 2020-08-16, 20:52 #951 sweety439     Nov 2016 23·103 Posts 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]
 2020-08-16, 20:54 #952 sweety439     Nov 2016 23×103 Posts Riesel: b=2: (see http://www.prothsearch.com/rieselprob.html) 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
 2020-08-16, 21:11 #953 sweety439     Nov 2016 23·103 Posts 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) Last fiddled with by sweety439 on 2020-08-16 at 21:14
2020-08-16, 21:33   #954
sweety439

Nov 2016

23·103 Posts

Quote:
 Originally Posted by sweety439 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)
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)

 2020-08-18, 01:55 #955 sweety439     Nov 2016 23·103 Posts https://github.com/xayahrainie4793/E...el-conjectures (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)) https://github.com/xayahrainie4793/f...el-conjectures (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) https://github.com/xayahrainie4793/all-k-1024 (2<=b<=32 or b=64, 256, 1<=k<=1024)
2020-08-18, 01:57   #956
sweety439

Nov 2016

23×103 Posts

Update zip file for 4<=b<=32 (b=2 and 3 are already in https://github.com/xayahrainie4793/E...el-conjectures) or b=64, 256, 1<=k<=1024, searched up to n=4096
Attached Files
 k le 1024.zip (177.3 KB, 12 views)

 2020-08-18, 19:45 #957 sweety439     Nov 2016 23·103 Posts Newest files: Sierpinski problems Riesel problems

 Similar Threads Thread Thread Starter Forum Replies Last Post sweety439 sweety439 11 2020-09-23 01:42 sweety439 sweety439 20 2020-07-03 17:22 sweety439 sweety439 12 2017-12-01 21:56 robert44444uk Conjectures 'R Us 139 2007-12-17 05:17 rogue Conjectures 'R Us 11 2007-12-17 05:08

All times are UTC. The time now is 17:32.

Thu Oct 29 17:32:11 UTC 2020 up 49 days, 14:43, 1 user, load averages: 2.16, 2.16, 2.14