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Old 2017-02-23, 09:50   #10
sweety439
 
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Nov 2016

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Quote:
Originally Posted by sweety439 View Post
If b and k are of these forms, then k is a Brier number (i.e. both Sierpinski number and Riesel number) to base b.

Code:
b               k
= 14 mod 15     = 4 or 11 mod 15
= 20 mod 21     = 8 or 13 mod 21
= 32 mod 33     = 10 or 23 mod 33
= 34 mod 35     = 6 or 29 mod 35
= 38 mod 39     = 14 or 25 mod 39
= 50 mod 51     = 16 or 35 mod 51
= 54 mod 55     = 21 or 34 mod 55
= 56 mod 57     = 20 or 37 mod 57
= 64 mod 65     = 14 or 51 mod 65
= 68 mod 69     = 22 or 47 mod 69
= 76 mod 77     = 34 or 43 mod 77
= 84 mod 85     = 16 or 69 mod 85
= 86 mod 87     = 28 or 59 mod 87
= 90 mod 91     = 27 or 64 mod 91
= 92 mod 93     = 32 or 61 mod 93
= 94 mod 95     = 39 or 56 mod 95
= 110 mod 111   = 38 or 73 mod 111
= 114 mod 115   = 24 or 91 mod 115
= 118 mod 119   = 50 or 69 mod 119
= 122 mod 123   = 40 or 83 mod 123
= 128 mod 129   = 44 or 85 mod 129
= 132 mod 133   = 20 or 113 mod 133
= 140 mod 141   = 46 or 95 mod 141
= 142 mod 143   = 12 or 131 mod 143
Generally, if there is a prime p divides both k+1 and b+1, and a prime q divides both k-1 and b+1, then k is both Sierpinski number (if gcd(k+1,b-1) = 1) and Riesel number (if gcd(k-1,b-1) = 1) to base b. (the covering set is both {p, q})

Thus, for the original Sierpinski/Riesel problems, if b+1 has at least two distinct odd prime factors, then it is easy to find a Sierpinski/Riesel k. Besides, for the reverse Sierpinski/Riesel problems, if neither k+1 nor k-1 is a power of 2 ("power of 2" includes 1), then it is easy to find a Sierpinski/Riesel base b.
Thus, for all Sierpinski/Riesel bases b<=144 such that b+1 has at least two distinct odd prime factors:

Code:
base      the smallest Sierpinski/Riesel k we calculated       the truly smallest Sierpinski/Riesel k
14        4                                                    both 4
20        8                                                    both 8
29        4                                                    both 4
32        10                                                   both 10
34        6                                                    both 6
38        14                                                   14 / 13 (13 is also a Riesel k to base 38)
41        8                                                    both 8
44        4                                                    both 4
50        16                                                   both 16
54        21                                                   both 21
56        20                                                   both 20
59        4                                                    both 4
62        8                                                    both 8
64        14                                                   51 / 14 (14 is a trivial k in the Sierpinski case)
65        10                                                   both 10
68        22                                                   both 22
69        6                                                    both 6
74        4                                                    both 4
76        34                                                   43 / 120 (34 is a trivial k in the Sierpinski case, and all of 34, 43 and 111 are trivial k's in the Riesel case)
77        14                                                   both 14
83        8                                                    both 8
84        16                                                   both 16
86        28                                                   both 28
89        4                                                    both 4
90        27                                                   both 27
92        32                                                   both 32
94        39                                                   both 39
98        10                                                   both 10
101       16                                                   16 / 118 (all of 16, 35, 67 and 86 are trivial k's in the Riesel case)
104       4                                                    both 4
109       21                                                   34 / 144 (21 is a trivial k in the Sierpinski case, and all of 21, 34, 76, 89 and 131 are trivial k's in the Riesel case)
110       38                                                   both 38
113       20                                                   94 / 20 (all of 20, 37 and 77 are trivial k's in the Sierpinski case)
114       24                                                   both 24
116       14                                                   25 / 14 (14 is a trivial k in the Sierpinski case)
118       50                                                   69 / 50 (50 is a trivial k in the Sierpinski case)
119       4                                                    both 4
122       40                                                   40 / 14 (14 is also a Riesel k to base 122)
125       8                                                    both 8
128       44                                                   both 44
129       14                                                   both 14
131       10                                                   both 10
132       20                                                   13 / 20 (13 is also a Sierpinski k to base 132)
134       4                                                    both 4
137       22                                                   both 22
139       6                                                    both 6
140       46                                                   both 46
142       12                                                   both 12
144       59                                                   both 59

Last fiddled with by sweety439 on 2017-02-23 at 16:27
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