## 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.

## Friday, November 29, 2013

### Theory intelligently applied: removing a Honda from the frozen Baikal

Teoria aplicata frumos scoate o Honda din Baikal :-)
Дубинушка или русская смекалка