Quote:
Originally Posted by sweety439
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