## Wednesday, May 08, 2019

### [lvlpjunt] Best random bytes

Here are some bytes, values in the range 0..255:

181 4 243 51 249 222 100 132 89 125 137 179 117 74 190 159 29 111 96 186 137 59 168 76 237 23 172 133 131 51 153 21 74 252 131 4 58 184 162 195 168 177 254 111 220 131 219 57 15 116 168 94 67 156 123 74 120 4 135 54 61 250 39 104 210 32 46 135 66 175 31 78 83 5 156 96 17 188 51 123 202 177 188 145 22 136 69 138 70 10 188 114 47 124 78 51 198 213 168 163 139 183 233 220 203 42 99 67 49 243 200 77 245 47 18 15 131 110 88 46 234 164 160 137 144 64 202 74 129 57 74 182 216 253 14 253 244 211 160 44 235 201 62 12 66 100 218 188 213 40 182 81 184 207 52 27 111 130 54 199 1 4 220 1 254 50 53 47 51 42 94 159 123 218 30 191 246 161 190 63 202 34 19 7 222 160 98 65 247 170 129 194 193 252 189 222 162 247 220 51 24 131 138 46 175 245 243 178 210 79 74 118 63 172 184 130 253 254 23 15 211 177 247 128 249 172 206 65 121 127 40 5 194 70 120 94 146 149 112 35 95 207 143 123 202 62 163 59 77 124 96 165 230 51 227 225

The bytes are the first 256 digits of sqrt(0.5) in base 256.  The first digit behaves as expected: 181^2 = 32761 ~= 2^15.  (Of course, one can compute many more bytes.)  We propose this sequence as a "gold standard" for randomness.  If you can find nonrandomness (especially non-normality) in these bytes, then it will have huge repercussions on other numbers like pi.

Digits 0 and 255 are 2 of the 104 digits that do not occur in the first 256 digits, but all digits do eventually occur, and are conjectured to occur equally often because of normality.

Is there a digit extraction algorithm (like BBP for pi) for this square root, allowing calculating a digit without having to calculate earlier digits?  If so, it would be a random number generator that one can fast forward.

Previously, writing various square roots in various bases.