Saturday, November 30, 2013

Small pairs of primitive roots with sum and difference one - raw data for primes up to 1013



Prime p
Primitive roots
Category 1
Sum 1
Small (least < \/p )
Category 2
Difference 1
Small (least < \/p )
2
1
no
no
3
2
no
no
5
2, 3
no
(2,3)
7
3, 5
no
no
11
2, 6, 7, 8
no
no
13
2, 6, 7, 11
no
no
17
3, 5, 6, 7, 10, 11, 12, 14
no
no
19
2, 3, 10, 13, 14, 15
no
(2,3)
23
5, 7, 10, 11, 14, 15, 17, 19, 20, 21
no
no
29
2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27
(3,27)
(2,3)
31
3, 11, 12, 13, 17, 21, 22, 24
no
no
37
2, 5, 13, 15, 17, 18, 19, 20, 22, 24, 32, 35
no
no
41
6, 7, 11, 12, 13, 15, 17, 19, 22, 24, 26, 28, 29, 30, 34, 35
no
no
43
3, 5, 12, 18, 19, 20, 26, 28, 29, 30, 33, 34
no
no
47
5, 10, 11, 13, 15, 19, 20, 22, 23, 26, 29, 30, 31, 33, 35, 38, 39, 40, 41, 43, 44, 45
(5,43)
no
53
2, 3, 5, 8, 12, 14, 18, 19, 20, 21, 22, 26, 27, 31, 32, 33, 34, 35, 39, 41, 45, 48, 50, 51
(3,51)
(2,3)


From now on, the set of all primitive roots (column 2 above) is not listed.



59
(6, 54)
no
61
(7, 55)
(6, 7)
67
(7, 61)
no
71
(7, 65)
no
73
no
no
79
(3, 77)
(6, 7)
83
(5, 79)
(5, 6)
89
(7, 83)
(6, 7)
97
no
no
101
(3, 99)
(2, 3)
103
(5, 99)
(5, 6)
107
(5, 103)
(5, 6)
109
no
(10, 11)
113
(6, 108)
(5, 6)
127
no
(6, 7)
131
(6, 126)
no
137
(6, 132)
(5, 6)
139
no
(2, 3)
149
(3, 147)
(2, 3)
151
(6, 146)
(6, 7)
157
(6, 152)
(5, 6)
163
(11, 153)
(11, 12)
167
(5, 163)
no
173
(3, 171)
(2, 3)
179
(6, 174)
(6, 7)
181
no
no
191
no
no
193
no
no
197
(3, 195)
(2, 3)
199
(3, 197)
no
211
(7, 205)
(2, 3)
223
(10, 214)
(5, 6)
227
(5, 223)
(13, 14)
229
(7, 223)
(6, 7)
233
(6, 228)
(5, 6)


Golomb construction


For the prime 47, and primitive roots 5 and 43 with sum 1 mod 47, the Golomb-type Costas array is defined as follows: a(I,J)=1 if 5^I+43^J=1 (mod 47), otherwise a(I,J)=0. 
MATLAB style (row #1 @ the bottom) - granted, an imperfect screen shot...
The displacement vectors between pairs of distinct points are all distinct, which makes it useful, for example, in radar design.