Talk:Find largest left truncatable prime in a given base
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Could someone make the example where it says "in base 10 candidates are 2,3,5,7. Putting a digit in the range 1 to base-1 in front of each candidate must result in a prime. So 2 and 5, like the whale and the petunias in The Hitchhiker's Guide to the Galaxy, come into existance only to be extinguished" clearer? I thought it was about 12 and 15, but it is not.
- Thanks. Can I remove this whole item? Or is it supposed to remain for some sort of historical documentation?
Hint for base 10, 12, 14 etc
Where the maximum left truncatable prime in a base is say 24 digits or more, to solve this task fully, try using a quick test for probable primality. Then only test for total compliance with your final candidate.--Nigel Galloway 11:50, 15 September 2012 (UTC)
Note, that setting the confidence too low will increase the number of composites which will need to be tested and so may not lead to optimal performance.
Using java.math.BigInteger.new(candidate).isProbablePrime(confidence):
- At 1 I found that the number of candidates at some levels increased over that achieved at 100.
- At 5 I have found the number of candidates at each level to be the same as the number found at 100 (where I have been able to determine the 100 number).
--Nigel Galloway 11:56, 21 September 2012 (UTC)
- It's fascinating that when I compare the number of primes at different stages using different levels of reliability testing, the difference with one Miller-Rabin round is only about 1% larger than the count with 5, and often rather less than that. This means that as the number of candidates narrows back down again as the prime size increases, the bad candidates seem to be being removed. Fascinating, that. Also, if you've not got a good MR implementation, you really need a decent
modpow()
function/operator; it makes a gigantic difference. (Now, to find a faster computer…) –Donal Fellows 09:09, 8 October 2012 (UTC)
- Yes, why is this task so easy? Richard G.E. Pinch records his personal research on strong psudoprimes here. Analysis of the data presented there indicates that the Largest Left Truncatable Prime algorithm is a bad generator of strong pseudoprimes, orders of magnitude worse than a random number generator. I have added a task here which introduces the subject to Rosetta Code.--Nigel Galloway 13:17, 30 November 2012 (UTC)
Number of left truncatable primes in a given base
You nay wish to keep a count of the number of left truncatable primes in a given base.--Nigel Galloway 11:50, 15 September 2012 (UTC)
Using the JRuby example to print the number of left truncatable primes by digit count produces:
1 1 1 The largest left truncatable prime in base 3 is 23 2 2 3 3 3 3 The largest left truncatable prime in base 4 is 4091 2 4 4 3 1 1 The largest left truncatable prime in base 5 is 7817 3 4 12 25 44 54 60 62 59 51 35 20 12 7 3 2 1 The largest left truncatable prime in base 6 is 4836525320399 3 6 6 4 1 1 1 The largest left truncatable prime in base 7 is 817337 4 12 29 50 66 77 61 51 38 27 17 8 3 2 1 The largest left truncatable prime in base 8 is 14005650767869 4 9 15 17 24 16 9 6 5 3 The largest left truncatable prime in base 9 is 1676456897 4 11 39 99 192 326 429 521 545 517 448 354 276 212 117 72 42 24 13 6 5 4 3 1 The largest left truncatable prime in base 10 is 357686312646216567629137 4 8 15 18 15 8 4 2 1 The largest left truncatable prime in base 11 is 2276005673 5 23 119 409 1124 2496 4733 7711 11231 14826 17341 18787 19001 17567 15169 12085 9272 6606 4451 2882 1796 1108 601 346 181 103 49 19 8 2 1 1 The largest left truncatable prime in base 12 is 13092430647736190817303130065827539 5 13 20 23 17 11 7 4 The largest left truncatable prime in base 13 is 812751503 6 26 101 300 678 1299 2093 3017 3751 4196 4197 3823 3206 2549 1908 1269 783 507 322 163 97 55 27 13 5 2 The largest left truncatable prime in base 14 is 615419590422100474355767356763 6 22 79 207 391 644 934 1177 1275 1167 1039 816 608 424 261 142 74 45 25 13 7 1 The largest left truncatable prime in base 15 is 34068645705927662447286191 6 31 124 337 749 1292 1973 2695 3210 3490 3335 2980 2525 1840 1278 878 556 326 174 93 50 25 9 5 1 The largest left truncatable prime in base 16 is 1088303707153521644968345559987 6 22 43 55 74 58 41 31 23 8 1 The largest left truncatable prime in base 17 is 13563641583101 7 49 311 1396 5117 15243 38818 85683 167132 293518 468456 680171 911723 1133959 1313343 1423597 1449405 1395514 1270222 1097353 902477 707896 529887 381239 2632 75 174684 111046 67969 40704 23201 12793 6722 3444 1714 859 422 205 98 39 14 7 1 1 The largest left truncatable prime in base 18 is 571933398724668544269594979167602382822769202133808087 7 29 61 106 122 117 93 66 36 18 10 10 6 4 The largest left truncatable prime in base 19 is 546207129080421139 8 56 321 1311 4156 10963 24589 47737 83011 129098 181707 234685 278792 306852 315113 302684 273080 233070 188331 145016 105557 73276 48819 31244 19237 11209 6209 3383 1689 917 430 196 80 44 20 7 2 The largest left truncatable prime in base 20 is 1073289911449776273800623217566610940096241078373 8 41 165 457 1079 2072 3316 4727 6003 6801 7051 6732 5862 4721 3505 2474 1662 1039 628 369 219 112 52 17 13 4 2 The largest left truncatable prime in base 21 is 391461911766647707547123429659688417 8 48 261 1035 3259 8247 17727 33302 55354 82379 111917 137857 158043 167442 165782 152997 132508 108083 83974 61950 43453 29212 18493 11352 6693 3738 2053 1125594 293 145 70 31 13 6 2 1 The largest left truncatable prime in base 22 is 33389741556593821170176571348673618833349516314271 8 30 78 137 181 200 186 171 121 100 67 41 24 16 9 2 1 The largest left truncatable prime in base 23 is 116516557991412919458949
--Nigel Galloway 12:41, 16 September 2012 (UTC)
Discrepancy in auxiliary output
Tcl vs...
The other solution showing these counts is Tcl and for base 12 it reports 1126 candidates at length 5 whereas the counts above for Perl 6 show 1124. This is the first discrepancy. Auxiliary output from the Fortran version also shows 1124 and matches the other counts from Perl 6 also. So, here are the 1124 five-digit base twelve candidates, in storage order:
Schedule
5 candidates at length 1 23 candidates at length 2 119 candidates at length 3 409 candidates at length 4 1124 candidates at length 5 1 241049 = 11.7.5.11.5 2 137369 = 6.7.5.11.5 3 75161 = 3.7.5.11.5 4 113177 = 5.5.5.11.5 5 50969 = 2.5.5.11.5 6 226937 = 10.11.3.11.5 7 164729 = 7.11.3.11.5 8 156089 = 7.6.3.11.5 9 135353 = 6.6.3.11.5 10 114617 = 5.6.3.11.5 11 216569 = 10.5.3.11.5 12 92153 = 4.5.3.11.5 13 235577 = 11.4.3.11.5 14 48953 = 2.4.3.11.5 15 230393 = 11.1.3.11.5 16 147449 = 7.1.3.11.5 17 126713 = 6.1.3.11.5 18 105977 = 5.1.3.11.5 19 226649 = 10.11.1.11.5 20 205913 = 9.11.1.11.5 21 185177 = 8.11.1.11.5 22 102233 = 4.11.1.11.5 23 60761 = 2.11.1.11.5 24 240473 = 11.7.1.11.5 25 53849 = 2.7.1.11.5 26 33113 = 1.7.1.11.5 27 235289 = 11.4.1.11.5 28 173081 = 8.4.1.11.5 29 69401 = 3.4.1.11.5 30 217697 = 10.5.11.9.5 31 196961 = 9.5.11.9.5 32 134753 = 6.5.11.9.5 33 93281 = 4.5.11.9.5 34 112289 = 5.4.11.9.5 35 25889 = 1.2.11.9.5 36 243473 = 11.8.10.9.5 37 202001 = 9.8.10.9.5 38 119057 = 5.8.10.9.5 39 98321 = 4.8.10.9.5 40 200273 = 9.7.10.9.5 41 117329 = 5.7.10.9.5 42 196817 = 9.5.10.9.5 43 176081 = 8.5.10.9.5 44 134609 = 6.5.10.9.5 45 234833 = 11.3.10.9.5 46 89681 = 4.3.10.9.5 47 212369 = 10.2.10.9.5 48 67217 = 3.2.10.9.5 49 118913 = 5.8.9.9.5 50 35969 = 1.8.9.9.5 51 239873 = 11.6.9.9.5 52 136193 = 6.6.9.9.5 53 23873 = 1.1.9.9.5 54 163409 = 7.10.6.9.5 55 142673 = 6.10.6.9.5 56 121937 = 5.10.6.9.5 57 59729 = 2.10.6.9.5 58 38993 = 1.10.6.9.5 59 182417 = 8.9.6.9.5 60 120209 = 5.9.6.9.5 61 78737 = 3.9.6.9.5 62 183569 = 8.10.2.9.5 63 142097 = 6.10.2.9.5 64 79889 = 3.10.2.9.5 65 157649 = 7.7.2.9.5 66 116177 = 5.7.2.9.5 67 95441 = 4.7.2.9.5 68 237137 = 11.5.2.9.5 69 216401 = 10.5.2.9.5 70 174929 = 8.5.2.9.5 71 71249 = 3.5.2.9.5 72 50513 = 2.5.2.9.5 73 188753 = 9.1.2.9.5 74 119489 = 5.9.1.9.5 75 78017 = 3.9.1.9.5 76 179969 = 8.8.1.9.5 77 159233 = 7.8.1.9.5 78 138497 = 6.8.1.9.5 79 76289 = 3.8.1.9.5 80 212801 = 10.3.1.9.5 81 171329 = 8.3.1.9.5 82 109121 = 5.3.1.9.5 83 26177 = 1.3.1.9.5 84 40853 = 1.11.7.8.5 85 182549 = 8.9.7.8.5 86 37397 = 1.9.7.8.5 87 92693 = 4.5.7.8.5 88 230933 = 11.1.7.8.5 89 85781 = 4.1.7.8.5 90 241013 = 11.7.5.8.5 91 175349 = 8.5.5.8.5 92 154613 = 7.5.5.8.5 93 133877 = 6.5.5.8.5 94 30197 = 1.5.5.8.5 95 44021 = 2.1.5.8.5 96 163109 = 7.10.4.8.5 97 121637 = 5.10.4.8.5 98 38693 = 1.10.4.8.5 99 223589 = 10.9.4.8.5 100 99173 = 4.9.4.8.5 101 78437 = 3.9.4.8.5 102 111269 = 5.4.4.8.5 103 90533 = 4.4.4.8.5 104 151013 = 7.3.4.8.5 105 109541 = 5.3.4.8.5 106 26597 = 1.3.4.8.5 107 211493 = 10.2.4.8.5 108 170021 = 8.2.4.8.5 109 128549 = 6.2.4.8.5 110 162821 = 7.10.2.8.5 111 121349 = 5.10.2.8.5 112 100613 = 4.10.2.8.5 113 59141 = 2.10.2.8.5 114 174917 = 8.5.2.8.5 115 154181 = 7.5.2.8.5 116 71237 = 3.5.2.8.5 117 233669 = 11.3.2.8.5 118 109253 = 5.3.2.8.5 119 26309 = 1.3.2.8.5 120 182537 = 8.9.7.7.5 121 78857 = 3.9.7.7.5 122 173897 = 8.4.7.7.5 123 49481 = 2.4.7.7.5 124 248057 = 11.11.6.7.5 125 185849 = 8.11.6.7.5 126 40697 = 1.11.6.7.5 127 246329 = 11.10.6.7.5 128 204857 = 9.10.6.7.5 129 199673 = 9.7.6.7.5 130 158201 = 7.7.6.7.5 131 54521 = 2.7.6.7.5 132 239417 = 11.6.6.7.5 133 218681 = 10.6.6.7.5 134 177209 = 8.6.6.7.5 135 115001 = 5.6.6.7.5 136 73529 = 3.6.6.7.5 137 32057 = 1.6.6.7.5 138 191033 = 9.2.6.7.5 139 149561 = 7.2.6.7.5 140 108089 = 5.2.6.7.5 141 66617 = 3.2.6.7.5 142 59561 = 2.10.5.7.5 143 244457 = 11.9.5.7.5 144 120041 = 5.9.5.7.5 145 78569 = 3.9.5.7.5 146 37097 = 1.9.5.7.5 147 242729 = 11.8.5.7.5 148 97577 = 4.8.5.7.5 149 239273 = 11.6.5.7.5 150 156329 = 7.6.5.7.5 151 135593 = 6.6.5.7.5 152 94121 = 4.6.5.7.5 153 99017 = 4.9.3.7.5 154 36809 = 1.9.3.7.5 155 218249 = 10.6.3.7.5 156 176777 = 8.6.3.7.5 157 156041 = 7.6.3.7.5 158 52361 = 2.6.3.7.5 159 237257 = 11.5.3.7.5 160 154313 = 7.5.3.7.5 161 194057 = 9.4.3.7.5 162 131849 = 6.4.3.7.5 163 159209 = 7.8.1.7.5 164 97001 = 4.8.1.7.5 165 55529 = 2.8.1.7.5 166 219689 = 10.7.1.7.5 167 198953 = 9.7.1.7.5 168 116009 = 5.7.1.7.5 169 95273 = 4.7.1.7.5 170 176489 = 8.6.1.7.5 171 135017 = 6.6.1.7.5 172 114281 = 5.6.1.7.5 173 31337 = 1.6.1.7.5 174 212777 = 10.3.1.7.5 175 109097 = 5.3.1.7.5 176 46889 = 2.3.1.7.5 177 26153 = 1.3.1.7.5 178 143333 = 6.10.11.4.5 179 122597 = 5.10.11.4.5 180 79397 = 3.9.11.4.5 181 58661 = 2.9.11.4.5 182 226133 = 10.10.10.4.5 183 205397 = 9.10.10.4.5 184 122453 = 5.10.10.4.5 185 39509 = 1.10.10.4.5 186 238229 = 11.5.10.4.5 187 176021 = 8.5.10.4.5 188 93077 = 4.5.10.4.5 189 72341 = 3.5.10.4.5 190 30869 = 1.5.10.4.5 191 231317 = 11.1.10.4.5 192 127637 = 6.1.10.4.5 193 23957 = 1.1.10.4.5 194 142757 = 6.10.7.4.5 195 122021 = 5.10.7.4.5 196 211877 = 10.2.7.4.5 197 191141 = 9.2.7.4.5 198 45989 = 2.2.7.4.5 199 25253 = 1.2.7.4.5 200 168677 = 8.1.7.4.5 201 85733 = 4.1.7.4.5 202 64997 = 3.1.7.4.5 203 237509 = 11.5.5.4.5 204 113093 = 5.5.5.4.5 205 92357 = 4.5.5.4.5 206 149381 = 7.2.5.4.5 207 204101 = 9.10.1.4.5 208 162629 = 7.10.1.4.5 209 121157 = 5.10.1.4.5 210 119429 = 5.9.1.4.5 211 57221 = 2.9.1.4.5 212 133253 = 6.5.1.4.5 213 91781 = 4.5.1.4.5 214 29573 = 1.5.1.4.5 215 214469 = 10.4.1.4.5 216 90053 = 4.4.1.4.5 217 69317 = 3.4.1.4.5 218 233477 = 11.3.1.4.5 219 150533 = 7.3.1.4.5 220 67589 = 3.3.1.4.5 221 46853 = 2.3.1.4.5 222 245129 = 11.9.10.3.5 223 203657 = 9.9.10.3.5 224 182921 = 8.9.10.3.5 225 120713 = 5.9.10.3.5 226 79241 = 3.9.10.3.5 227 200201 = 9.7.10.3.5 228 137993 = 6.7.10.3.5 229 55049 = 2.7.10.3.5 230 34313 = 1.7.10.3.5 231 212297 = 10.2.10.3.5 232 191561 = 9.2.10.3.5 233 150089 = 7.2.10.3.5 234 87881 = 4.2.10.3.5 235 25673 = 1.2.10.3.5 236 169097 = 8.1.10.3.5 237 148361 = 7.1.10.3.5 238 227561 = 10.11.8.3.5 239 61673 = 2.11.8.3.5 240 160169 = 7.8.8.3.5 241 97961 = 4.8.8.3.5 242 56489 = 2.8.8.3.5 243 35753 = 1.8.8.3.5 244 237929 = 11.5.8.3.5 245 113513 = 5.5.8.3.5 246 170537 = 8.2.8.3.5 247 225689 = 10.10.7.3.5 248 163481 = 7.10.7.3.5 249 101273 = 4.10.7.3.5 250 80537 = 3.10.7.3.5 251 154841 = 7.5.7.3.5 252 230873 = 11.1.7.3.5 253 189401 = 9.1.7.3.5 254 44249 = 2.1.7.3.5 255 163193 = 7.10.5.3.5 256 121721 = 5.10.5.3.5 257 59513 = 2.10.5.3.5 258 173561 = 8.4.5.3.5 259 90617 = 4.4.5.3.5 260 28409 = 1.4.5.3.5 261 107897 = 5.2.5.3.5 262 24953 = 1.2.5.3.5 263 199337 = 9.7.4.3.5 264 178601 = 8.7.4.3.5 265 216617 = 10.5.4.3.5 266 154409 = 7.5.4.3.5 267 133673 = 6.5.4.3.5 268 152681 = 7.4.4.3.5 269 90473 = 4.4.4.3.5 270 69737 = 3.4.4.3.5 271 109481 = 5.3.4.3.5 272 211433 = 10.2.4.3.5 273 128489 = 6.2.4.3.5 274 24809 = 1.2.4.3.5 275 221069 = 10.7.11.2.5 276 117389 = 5.7.11.2.5 277 157133 = 7.6.11.2.5 278 136397 = 6.6.11.2.5 279 74189 = 3.6.11.2.5 280 53453 = 2.6.11.2.5 281 32717 = 1.6.11.2.5 282 134669 = 6.5.11.2.5 283 113933 = 5.5.11.2.5 284 72461 = 3.5.11.2.5 285 246557 = 11.10.8.2.5 286 225821 = 10.10.8.2.5 287 163613 = 7.10.8.2.5 288 80669 = 3.10.8.2.5 289 198173 = 9.6.8.2.5 290 73757 = 3.6.8.2.5 291 194717 = 9.4.8.2.5 292 173981 = 8.4.8.2.5 293 111773 = 5.4.8.2.5 294 211997 = 10.2.8.2.5 295 25373 = 1.2.8.2.5 296 148061 = 7.1.8.2.5 297 85853 = 4.1.8.2.5 298 44381 = 2.1.8.2.5 299 164621 = 7.11.3.2.5 300 81677 = 3.11.3.2.5 301 232013 = 11.2.3.2.5 302 149069 = 7.2.3.2.5 303 86861 = 4.2.3.2.5 304 45389 = 2.2.3.2.5 305 162749 = 7.10.2.2.5 306 59069 = 2.10.2.2.5 307 38333 = 1.10.2.2.5 308 240509 = 11.7.2.2.5 309 199037 = 9.7.2.2.5 310 178301 = 8.7.2.2.5 311 33149 = 1.7.2.2.5 312 214589 = 10.4.2.2.5 313 152381 = 7.4.2.2.5 314 110909 = 5.4.2.2.5 315 90173 = 4.4.2.2.5 316 129917 = 6.3.2.2.5 317 67709 = 3.3.2.2.5 318 26237 = 1.3.2.2.5 319 185069 = 8.11.1.2.5 320 122861 = 5.11.1.2.5 321 242093 = 11.8.1.2.5 322 55469 = 2.8.1.2.5 323 219629 = 10.7.1.2.5 324 95213 = 4.7.1.2.5 325 126317 = 6.1.1.2.5 326 64109 = 3.1.1.2.5 327 22637 = 1.1.1.2.5 328 241793 = 11.7.11.1.5 329 158849 = 7.7.11.1.5 330 138113 = 6.7.11.1.5 331 198593 = 9.6.11.1.5 332 74177 = 3.6.11.1.5 333 53441 = 2.6.11.1.5 334 236609 = 11.4.11.1.5 335 195137 = 9.4.11.1.5 336 132929 = 6.4.11.1.5 337 91457 = 4.4.11.1.5 338 169217 = 8.1.11.1.5 339 65537 = 3.1.11.1.5 340 165041 = 7.11.6.1.5 341 244529 = 11.9.6.1.5 342 203057 = 9.9.6.1.5 343 99377 = 4.9.6.1.5 344 50993 = 2.5.6.1.5 345 235889 = 11.4.6.1.5 346 215153 = 10.4.6.1.5 347 70001 = 3.4.6.1.5 348 232433 = 11.2.6.1.5 349 149489 = 7.2.6.1.5 350 87281 = 4.2.6.1.5 351 25073 = 1.2.6.1.5 352 147761 = 7.1.6.1.5 353 64817 = 3.1.6.1.5 354 199313 = 9.7.4.1.5 355 157841 = 7.7.4.1.5 356 95633 = 4.7.4.1.5 357 74897 = 3.7.4.1.5 358 169937 = 8.2.4.1.5 359 86993 = 4.2.4.1.5 360 247553 = 11.11.3.1.5 361 226817 = 10.11.3.1.5 362 206081 = 9.11.3.1.5 363 143873 = 6.11.3.1.5 364 40193 = 1.11.3.1.5 365 225089 = 10.10.3.1.5 366 204353 = 9.10.3.1.5 367 162881 = 7.10.3.1.5 368 100673 = 4.10.3.1.5 369 223361 = 10.9.3.1.5 370 181889 = 8.9.3.1.5 371 140417 = 6.9.3.1.5 372 103451 = 4.11.10.4.11 373 61979 = 2.11.10.4.11 374 41243 = 1.11.10.4.11 375 239963 = 11.6.10.4.11 376 198491 = 9.6.10.4.11 377 157019 = 7.6.10.4.11 378 115547 = 5.6.10.4.11 379 94811 = 4.6.10.4.11 380 32603 = 1.6.10.4.11 381 217499 = 10.5.10.4.11 382 155291 = 7.5.10.4.11 383 113819 = 5.5.10.4.11 384 93083 = 4.5.10.4.11 385 176747 = 8.6.3.4.11 386 156011 = 7.6.3.4.11 387 149099 = 7.2.3.4.11 388 24683 = 1.2.3.4.11 389 245723 = 11.10.2.4.11 390 204251 = 9.10.2.4.11 391 162779 = 7.10.2.4.11 392 109211 = 5.3.2.4.11 393 26267 = 1.3.2.4.11 394 169691 = 8.2.2.4.11 395 238247 = 11.5.10.5.11 396 155303 = 7.5.10.5.11 397 236519 = 11.4.10.5.11 398 195047 = 9.4.10.5.11 399 174311 = 8.4.10.5.11 400 112103 = 5.4.10.5.11 401 91367 = 4.4.10.5.11 402 87911 = 4.2.10.5.11 403 46439 = 2.2.10.5.11 404 25703 = 1.2.10.5.11 405 241559 = 11.7.9.5.11 406 200087 = 9.7.9.5.11 407 179351 = 8.7.9.5.11 408 238103 = 11.5.9.5.11 409 217367 = 10.5.9.5.11 410 82759 = 3.11.10.8.7 411 124231 = 5.11.10.8.7 412 144967 = 6.11.10.8.7 413 165703 = 7.11.10.8.7 414 46471 = 2.2.10.8.7 415 87943 = 4.2.10.8.7 416 150151 = 7.2.10.8.7 417 170887 = 8.2.10.8.7 418 24007 = 1.1.10.8.7 419 65479 = 3.1.10.8.7 420 169159 = 8.1.10.8.7 421 210631 = 10.1.10.8.7 422 231367 = 11.1.10.8.7 423 41143 = 1.11.9.8.7 424 61879 = 2.11.9.8.7 425 124087 = 5.11.9.8.7 426 165559 = 7.11.9.8.7 427 184567 = 8.10.9.8.7 428 51511 = 2.5.9.8.7 429 113719 = 5.5.9.8.7 430 155191 = 7.5.9.8.7 431 196663 = 9.5.9.8.7 432 49783 = 2.4.9.8.7 433 236407 = 11.4.9.8.7 434 57991 = 2.9.6.8.7 435 120199 = 5.9.6.8.7 436 178951 = 8.7.6.8.7 437 199687 = 9.7.6.8.7 438 90823 = 4.4.6.8.7 439 215239 = 10.4.6.8.7 440 47623 = 2.3.6.8.7 441 109831 = 5.3.6.8.7 442 151303 = 7.3.6.8.7 443 191047 = 9.2.6.8.7 444 158071 = 7.7.5.8.7 445 178807 = 8.7.5.8.7 446 220279 = 10.7.5.8.7 447 71671 = 3.5.5.8.7 448 113143 = 5.5.5.8.7 449 196087 = 9.5.5.8.7 450 88951 = 4.3.5.8.7 451 130423 = 6.3.5.8.7 452 192631 = 9.3.5.8.7 453 234103 = 11.3.5.8.7 454 126967 = 6.1.5.8.7 455 147703 = 7.1.5.8.7 456 230647 = 11.1.5.8.7 457 75967 = 3.7.11.6.7 458 96703 = 4.7.11.6.7 459 200383 = 9.7.11.6.7 460 31039 = 1.5.11.6.7 461 113983 = 5.5.11.6.7 462 176191 = 8.5.11.6.7 463 196927 = 9.5.11.6.7 464 46591 = 2.2.11.6.7 465 108799 = 5.2.11.6.7 466 171007 = 8.2.11.6.7 467 212479 = 10.2.11.6.7 468 65599 = 3.1.11.6.7 469 107071 = 5.1.11.6.7 470 127807 = 6.1.11.6.7 471 54799 = 2.7.8.6.7 472 137743 = 6.7.8.6.7 473 220687 = 10.7.8.6.7 474 51343 = 2.5.8.6.7 475 134287 = 6.5.8.6.7 476 175759 = 8.5.8.6.7 477 237967 = 11.5.8.6.7 478 39103 = 1.10.7.6.7 479 75391 = 3.7.7.6.7 480 199807 = 9.7.7.6.7 481 220543 = 10.7.7.6.7 482 47743 = 2.3.7.6.7 483 130687 = 6.3.7.6.7 484 151423 = 7.3.7.6.7 485 161071 = 7.9.2.6.7 486 91951 = 4.5.2.6.7 487 112687 = 5.5.2.6.7 488 154159 = 7.5.2.6.7 489 86767 = 4.2.2.6.7 490 128239 = 6.2.2.6.7 491 231919 = 11.2.2.6.7 492 60703 = 2.11.1.6.7 493 81439 = 3.11.1.6.7 494 29599 = 1.5.1.6.7 495 91807 = 4.5.1.6.7 496 112543 = 5.5.1.6.7 497 133279 = 6.5.1.6.7 498 150559 = 7.3.1.6.7 499 209311 = 10.1.1.6.7 500 230047 = 11.1.1.6.7 501 49171 = 2.4.5.5.7 502 152851 = 7.4.5.5.7 503 194323 = 9.4.5.5.7 504 24979 = 1.2.5.5.7 505 87187 = 4.2.5.5.7 506 107923 = 5.2.5.5.7 507 128659 = 6.2.5.5.7 508 178627 = 8.7.4.5.7 509 92227 = 4.5.4.5.7 510 195907 = 9.5.4.5.7 511 237379 = 11.5.4.5.7 512 88771 = 4.3.4.5.7 513 109507 = 5.3.4.5.7 514 150979 = 7.3.4.5.7 515 233923 = 11.3.4.5.7 516 149251 = 7.2.4.5.7 517 169987 = 8.2.4.5.7 518 64579 = 3.1.4.5.7 519 230467 = 11.1.4.5.7 520 102451 = 4.11.3.5.7 521 247603 = 11.11.3.5.7 522 35059 = 1.8.3.5.7 523 138739 = 6.8.3.5.7 524 180211 = 8.8.3.5.7 525 242419 = 11.8.3.5.7 526 93811 = 4.6.3.5.7 527 114547 = 5.6.3.5.7 528 135283 = 6.6.3.5.7 529 156019 = 7.6.3.5.7 530 218227 = 10.6.3.5.7 531 48883 = 2.4.3.5.7 532 111091 = 5.4.3.5.7 533 152563 = 7.4.3.5.7 534 214771 = 10.4.3.5.7 535 67891 = 3.3.3.5.7 536 109363 = 5.3.3.5.7 537 130099 = 6.3.3.5.7 538 171571 = 8.3.3.5.7 539 192307 = 9.3.3.5.7 540 213043 = 10.3.3.5.7 541 164371 = 7.11.1.5.7 542 58963 = 2.10.1.5.7 543 79699 = 3.10.1.5.7 544 121171 = 5.10.1.5.7 545 141907 = 6.10.1.5.7 546 245587 = 11.10.1.5.7 547 36187 = 1.8.11.3.7 548 56923 = 2.8.11.3.7 549 77659 = 3.8.11.3.7 550 119131 = 5.8.11.3.7 551 160603 = 7.8.11.3.7 552 75931 = 3.7.11.3.7 553 96667 = 4.7.11.3.7 554 138139 = 6.7.11.3.7 555 221083 = 10.7.11.3.7 556 113947 = 5.5.11.3.7 557 134683 = 6.5.11.3.7 558 238363 = 11.5.11.3.7 559 195163 = 9.4.11.3.7 560 215899 = 10.4.11.3.7 561 56779 = 2.8.10.3.7 562 98251 = 4.8.10.3.7 563 243403 = 11.8.10.3.7 564 37339 = 1.9.7.3.7 565 120283 = 5.9.7.3.7 566 203227 = 9.9.7.3.7 567 223963 = 10.9.7.3.7 568 116827 = 5.7.7.3.7 569 90907 = 4.4.7.3.7 570 173851 = 8.4.7.3.7 571 25243 = 1.2.7.3.7 572 45979 = 2.2.7.3.7 573 108187 = 5.2.7.3.7 574 128923 = 6.2.7.3.7 575 211867 = 10.2.7.3.7 576 118411 = 5.8.6.3.7 577 33739 = 1.7.6.3.7 578 75211 = 3.7.6.3.7 579 95947 = 4.7.6.3.7 580 113227 = 5.5.6.3.7 581 133963 = 6.5.6.3.7 582 154699 = 7.5.6.3.7 583 196171 = 9.5.6.3.7 584 47563 = 2.3.6.3.7 585 151243 = 7.3.6.3.7 586 234187 = 11.3.6.3.7 587 66571 = 3.2.6.3.7 588 211723 = 10.2.6.3.7 589 232459 = 11.2.6.3.7 590 31723 = 1.6.4.3.7 591 135403 = 6.6.4.3.7 592 156139 = 7.6.4.3.7 593 209707 = 10.1.4.3.7 594 133387 = 6.5.2.3.7 595 174859 = 8.5.2.3.7 596 237067 = 11.5.2.3.7 597 90187 = 4.4.2.3.7 598 110923 = 5.4.2.3.7 599 214603 = 10.4.2.3.7 600 103423 = 4.11.10.2.7 601 34303 = 1.7.10.2.7 602 137983 = 6.7.10.2.7 603 200191 = 9.7.10.2.7 604 241663 = 11.7.10.2.7 605 74047 = 3.6.10.2.7 606 198463 = 9.6.10.2.7 607 238207 = 11.5.10.2.7 608 79087 = 3.9.9.2.7 609 99823 = 4.9.9.2.7 610 224239 = 10.9.9.2.7 611 30703 = 1.5.9.2.7 612 51439 = 2.5.9.2.7 613 113647 = 5.5.9.2.7 614 155119 = 7.5.9.2.7 615 108463 = 5.2.9.2.7 616 61519 = 2.11.7.2.7 617 123727 = 5.11.7.2.7 618 227407 = 10.11.7.2.7 619 51151 = 2.5.7.2.7 620 71887 = 3.5.7.2.7 621 92623 = 4.5.7.2.7 622 113359 = 5.5.7.2.7 623 196303 = 9.5.7.2.7 624 28687 = 1.4.7.2.7 625 132367 = 6.4.7.2.7 626 173839 = 8.4.7.2.7 627 85711 = 4.1.7.2.7 628 147919 = 7.1.7.2.7 629 189391 = 9.1.7.2.7 630 210127 = 10.1.7.2.7 631 230863 = 11.1.7.2.7 632 61231 = 2.11.5.2.7 633 81967 = 3.11.5.2.7 634 123439 = 5.11.5.2.7 635 164911 = 7.11.5.2.7 636 206383 = 9.11.5.2.7 637 73327 = 3.6.5.2.7 638 94063 = 4.6.5.2.7 639 114799 = 5.6.5.2.7 640 177007 = 8.6.5.2.7 641 218479 = 10.6.5.2.7 642 154543 = 7.5.5.2.7 643 216751 = 10.5.5.2.7 644 237487 = 11.5.5.2.7 645 55903 = 2.8.4.2.7 646 137119 = 6.7.4.2.7 647 220063 = 10.7.4.2.7 648 109471 = 5.3.4.2.7 649 171679 = 8.3.4.2.7 650 60943 = 2.11.3.2.7 651 164623 = 7.11.3.2.7 652 185359 = 8.11.3.2.7 653 31567 = 1.6.3.2.7 654 73039 = 3.6.3.2.7 655 218191 = 10.6.3.2.7 656 133519 = 6.5.3.2.7 657 174991 = 8.5.3.2.7 658 47119 = 2.3.3.2.7 659 88591 = 4.3.3.2.7 660 192271 = 9.3.3.2.7 661 233743 = 11.3.3.2.7 662 107599 = 5.2.3.2.7 663 190543 = 9.2.3.2.7 664 37747 = 1.9.10.1.7 665 120691 = 5.9.10.1.7 666 182899 = 8.9.10.1.7 667 245107 = 11.9.10.1.7 668 32563 = 1.6.10.1.7 669 53299 = 2.6.10.1.7 670 94771 = 4.6.10.1.7 671 156979 = 7.6.10.1.7 672 219187 = 10.6.10.1.7 673 72307 = 3.5.10.1.7 674 113779 = 5.5.10.1.7 675 155251 = 7.5.10.1.7 676 86131 = 4.1.10.1.7 677 106867 = 5.1.10.1.7 678 148339 = 7.1.10.1.7 679 122131 = 5.10.8.1.7 680 142867 = 6.10.8.1.7 681 96211 = 4.7.8.1.7 682 158419 = 7.7.8.1.7 683 241363 = 11.7.8.1.7 684 51283 = 2.5.8.1.7 685 72019 = 3.5.8.1.7 686 134227 = 6.5.8.1.7 687 175699 = 8.5.8.1.7 688 40627 = 1.11.6.1.7 689 61363 = 2.11.6.1.7 690 144307 = 6.11.6.1.7 691 227251 = 10.11.6.1.7 692 95923 = 4.7.6.1.7 693 199603 = 9.7.6.1.7 694 73459 = 3.6.6.1.7 695 218611 = 10.6.6.1.7 696 239347 = 11.6.6.1.7 697 130483 = 6.3.6.1.7 698 149491 = 7.2.6.1.7 699 170227 = 8.2.6.1.7 700 76771 = 3.8.5.1.7 701 242659 = 11.8.5.1.7 702 88867 = 4.3.5.1.7 703 171811 = 8.3.5.1.7 704 192547 = 9.3.5.1.7 705 23203 = 1.1.5.1.7 706 85411 = 4.1.5.1.7 707 230563 = 11.1.5.1.7 708 57331 = 2.9.2.1.7 709 223219 = 10.9.2.1.7 710 69427 = 3.4.2.1.7 711 90163 = 4.4.2.1.7 712 110899 = 5.4.2.1.7 713 121123 = 5.10.1.1.7 714 204067 = 9.10.1.1.7 715 31267 = 1.6.1.1.7 716 72739 = 3.6.1.1.7 717 134947 = 6.6.1.1.7 718 176419 = 8.6.1.1.7 719 238627 = 11.6.1.1.7 720 71011 = 3.5.1.1.7 721 195427 = 9.5.1.1.7 722 26083 = 1.3.1.1.7 723 46819 = 2.3.1.1.7 724 129763 = 6.3.1.1.7 725 65657 = 3.1.11.11.5 726 210809 = 10.1.11.11.5 727 41177 = 1.11.9.11.5 728 124121 = 5.11.9.11.5 729 248537 = 11.11.9.11.5 730 122393 = 5.10.9.11.5 731 246809 = 11.10.9.11.5 732 32537 = 1.6.9.11.5 733 136217 = 6.6.9.11.5 734 87833 = 4.2.9.11.5 735 150041 = 7.2.9.11.5 736 170777 = 8.2.9.11.5 737 165449 = 7.11.8.11.5 738 53129 = 2.6.8.11.5 739 115337 = 5.6.8.11.5 740 198281 = 9.6.8.11.5 741 219017 = 10.6.8.11.5 742 239753 = 11.6.8.11.5 743 155081 = 7.5.8.11.5 744 89417 = 4.3.8.11.5 745 213833 = 10.3.8.11.5 746 25679 = 1.2.10.3.11 747 87887 = 4.2.10.3.11 748 28559 = 1.4.6.3.11 749 215183 = 10.4.6.3.11 750 235919 = 11.4.6.3.11 751 133967 = 6.5.6.3.11 752 216911 = 10.5.6.3.11 753 177167 = 8.6.6.3.11 754 116687 = 5.7.6.3.11 755 59663 = 2.10.6.3.11 756 142607 = 6.10.6.3.11 757 206543 = 9.11.6.3.11 758 22943 = 1.1.3.3.11 759 188831 = 9.1.3.3.11 760 209567 = 10.1.3.3.11 761 230303 = 11.1.3.3.11 762 31583 = 1.6.3.3.11 763 238943 = 11.6.3.3.11 764 76511 = 3.8.3.3.11 765 200927 = 9.8.3.3.11 766 242399 = 11.8.3.3.11 767 131519 = 6.4.1.3.11 768 193727 = 9.4.1.3.11 769 214463 = 10.4.1.3.11 770 235199 = 11.4.1.3.11 771 29567 = 1.5.1.3.11 772 71039 = 3.5.1.3.11 773 72767 = 3.6.1.3.11 774 93503 = 4.6.1.3.11 775 58943 = 2.10.1.3.11 776 121151 = 5.10.1.3.11 777 162623 = 7.10.1.3.11 778 224831 = 10.10.1.3.11 779 27527 = 1.3.11.1.11 780 30983 = 1.5.11.1.11 781 51719 = 2.5.11.1.11 782 155399 = 7.5.11.1.11 783 196871 = 9.5.11.1.11 784 58631 = 2.9.11.1.11 785 79367 = 3.9.11.1.11 786 100103 = 4.9.11.1.11 787 183047 = 8.9.11.1.11 788 70439 = 3.4.9.1.11 789 132647 = 6.4.9.1.11 790 37607 = 1.9.9.1.11 791 120551 = 5.9.9.1.11 792 165479 = 7.11.9.1.11 793 206951 = 9.11.9.1.11 794 73607 = 3.6.7.1.11 795 94343 = 4.6.7.1.11 796 115079 = 5.6.7.1.11 797 33863 = 1.7.7.1.11 798 199751 = 9.7.7.1.11 799 39047 = 1.10.7.1.11 800 184199 = 8.10.7.1.11 801 225671 = 10.10.7.1.11 802 116663 = 5.7.6.1.11 803 137399 = 6.7.6.1.11 804 241079 = 11.7.6.1.11 805 130343 = 6.3.5.1.11 806 213287 = 10.3.5.1.11 807 40487 = 1.11.5.1.11 808 61223 = 2.11.5.1.11 809 144167 = 6.11.5.1.11 810 227111 = 10.11.5.1.11 811 247847 = 11.11.5.1.11 812 130199 = 6.3.4.1.11 813 171671 = 8.3.4.1.11 814 192407 = 9.3.4.1.11 815 233879 = 11.3.4.1.11 816 112919 = 5.5.4.1.11 817 195863 = 9.5.4.1.11 818 35159 = 1.8.4.1.11 819 76631 = 3.8.4.1.11 820 97367 = 4.8.4.1.11 821 180311 = 8.8.4.1.11 822 242519 = 11.8.4.1.11 823 86711 = 4.2.2.1.11 824 190391 = 9.2.2.1.11 825 50423 = 2.5.2.1.11 826 81527 = 3.11.2.1.11 827 164471 = 7.11.2.1.11 828 24359 = 1.2.1.1.11 829 65831 = 3.2.1.1.11 830 231719 = 11.2.1.1.11 831 67559 = 3.3.1.1.11 832 150503 = 7.3.1.1.11 833 90023 = 4.4.1.1.11 834 152231 = 7.4.1.1.11 835 193703 = 9.4.1.1.11 836 214439 = 10.4.1.1.11 837 32999 = 1.7.1.1.11 838 74471 = 3.7.1.1.11 839 178151 = 8.7.1.1.11 840 191803 = 9.2.11.11.7 841 110587 = 5.3.11.11.7 842 235003 = 11.3.11.11.7 843 41467 = 1.11.11.11.7 844 82939 = 3.11.11.11.7 845 165883 = 7.11.11.11.7 846 186619 = 8.11.11.11.7 847 248827 = 11.11.11.11.7 848 24043 = 1.1.10.11.7 849 193387 = 9.3.10.11.7 850 60331 = 2.10.10.11.7 851 164011 = 7.10.10.11.7 852 205483 = 9.10.10.11.7 853 44491 = 2.1.8.11.7 854 106699 = 5.1.8.11.7 855 148171 = 7.1.8.11.7 856 189643 = 9.1.8.11.7 857 72139 = 3.5.8.11.7 858 155083 = 7.5.8.11.7 859 37579 = 1.9.8.11.7 860 99787 = 4.9.8.11.7 861 244939 = 11.9.8.11.7 862 192667 = 9.3.5.11.7 863 234139 = 11.3.5.11.7 864 35419 = 1.8.5.11.7 865 201307 = 9.8.5.11.7 866 222043 = 10.8.5.11.7 867 242779 = 11.8.5.11.7 868 120091 = 5.9.5.11.7 869 140827 = 6.9.5.11.7 870 161563 = 7.9.5.11.7 871 244507 = 11.9.5.11.7 872 61339 = 2.11.5.11.7 873 102811 = 4.11.5.11.7 874 123547 = 5.11.5.11.7 875 24763 = 1.2.3.11.7 876 121531 = 5.10.3.11.7 877 163003 = 7.10.3.11.7 878 105691 = 5.1.1.11.7 879 147163 = 7.1.1.11.7 880 167899 = 8.1.1.11.7 881 209371 = 10.1.1.11.7 882 230107 = 11.1.1.11.7 883 148891 = 7.2.1.11.7 884 169627 = 8.2.1.11.7 885 91867 = 4.5.1.11.7 886 112603 = 5.5.1.11.7 887 237019 = 11.5.1.11.7 888 31387 = 1.6.1.11.7 889 72859 = 3.6.1.11.7 890 238747 = 11.6.1.11.7 891 74587 = 3.7.1.11.7 892 34843 = 1.8.1.11.7 893 55579 = 2.8.1.11.7 894 117787 = 5.8.1.11.7 895 200731 = 9.8.1.11.7 896 25759 = 1.2.10.10.7 897 67231 = 3.2.10.10.7 898 129439 = 6.2.10.10.7 899 212383 = 10.2.10.10.7 900 243487 = 11.8.10.10.7 901 48079 = 2.3.9.10.7 902 131023 = 6.3.9.10.7 903 144847 = 6.11.9.10.7 904 28927 = 1.4.8.10.7 905 49663 = 2.4.8.10.7 906 111871 = 5.4.8.10.7 907 132607 = 6.4.8.10.7 908 153343 = 7.4.8.10.7 909 174079 = 8.4.8.10.7 910 236287 = 11.4.8.10.7 911 37567 = 1.9.8.10.7 912 79039 = 3.9.8.10.7 913 120511 = 5.9.8.10.7 914 161983 = 7.9.8.10.7 915 41023 = 1.11.8.10.7 916 103231 = 4.11.8.10.7 917 206911 = 9.11.8.10.7 918 64927 = 3.1.6.10.7 919 25183 = 1.2.6.10.7 920 108127 = 5.2.6.10.7 921 191071 = 9.2.6.10.7 922 30367 = 1.5.6.10.7 923 134047 = 6.5.6.10.7 924 175519 = 8.5.6.10.7 925 216991 = 10.5.6.10.7 926 147151 = 7.1.1.10.7 927 167887 = 8.1.1.10.7 928 209359 = 10.1.1.10.7 929 231823 = 11.2.1.10.7 930 114319 = 5.6.1.10.7 931 36559 = 1.9.1.10.7 932 78031 = 3.9.1.10.7 933 119503 = 5.9.1.10.7 934 181711 = 8.9.1.10.7 935 38287 = 1.10.1.10.7 936 59023 = 2.10.1.10.7 937 162703 = 7.10.1.10.7 938 183439 = 8.10.1.10.7 939 224911 = 10.10.1.10.7 940 69539 = 3.4.2.10.11 941 173219 = 8.4.2.10.11 942 214691 = 10.4.2.10.11 943 92003 = 4.5.2.10.11 944 154211 = 7.5.2.10.11 945 97187 = 4.8.2.10.11 946 200867 = 9.8.2.10.11 947 221603 = 10.8.2.10.11 948 78179 = 3.9.2.10.11 949 161123 = 7.9.2.10.11 950 223331 = 10.9.2.10.11 951 40163 = 1.11.2.10.11 952 60899 = 2.11.2.10.11 953 206051 = 9.11.2.10.11 954 45491 = 2.2.3.10.11 955 107699 = 5.2.3.10.11 956 71411 = 3.5.3.10.11 957 95603 = 4.7.3.10.11 958 220019 = 10.7.3.10.11 959 159539 = 7.8.3.10.11 960 201011 = 9.8.3.10.11 961 221747 = 10.8.3.10.11 962 242483 = 11.8.3.10.11 963 61043 = 2.11.3.10.11 964 23747 = 1.1.8.10.11 965 44483 = 2.1.8.10.11 966 168899 = 8.1.8.10.11 967 231107 = 11.1.8.10.11 968 92867 = 4.5.8.10.11 969 134339 = 6.5.8.10.11 970 175811 = 8.5.8.10.11 971 238019 = 11.5.8.10.11 972 118787 = 5.8.8.10.11 973 201731 = 9.8.8.10.11 974 243203 = 11.8.8.10.11 975 30803 = 1.5.9.10.11 976 51539 = 2.5.9.10.11 977 155219 = 7.5.9.10.11 978 238163 = 11.5.9.10.11 979 34259 = 1.7.9.10.11 980 75731 = 3.7.9.10.11 981 117203 = 5.7.9.10.11 982 179411 = 8.7.9.10.11 983 58451 = 2.9.9.10.11 984 79187 = 3.9.9.10.11 985 99923 = 4.9.9.10.11 986 182867 = 8.9.9.10.11 987 44771 = 2.1.10.10.11 988 86243 = 4.1.10.10.11 989 106979 = 5.1.10.10.11 990 210659 = 10.1.10.10.11 991 51683 = 2.5.10.10.11 992 113891 = 5.5.10.10.11 993 238307 = 11.5.10.10.11 994 53411 = 2.6.10.10.11 995 141539 = 6.9.10.10.11 996 43451 = 2.1.1.8.11 997 64187 = 3.1.1.8.11 998 188603 = 9.1.1.8.11 999 33083 = 1.7.1.8.11 1000 53819 = 2.7.1.8.11 1001 116027 = 5.7.1.8.11 1002 198971 = 9.7.1.8.11 1003 219707 = 10.7.1.8.11 1004 102203 = 4.11.1.8.11 1005 122939 = 5.11.1.8.11 1006 205883 = 9.11.1.8.11 1007 126827 = 6.1.4.8.11 1008 209771 = 10.1.4.8.11 1009 230507 = 11.1.4.8.11 1010 66347 = 3.2.4.8.11 1011 87083 = 4.2.4.8.11 1012 190763 = 9.2.4.8.11 1013 211499 = 10.2.4.8.11 1014 47339 = 2.3.4.8.11 1015 88811 = 4.3.4.8.11 1016 109547 = 5.3.4.8.11 1017 30059 = 1.5.4.8.11 1018 175211 = 8.5.4.8.11 1019 66491 = 3.2.5.8.11 1020 211643 = 10.2.5.8.11 1021 68219 = 3.3.5.8.11 1022 151163 = 7.3.5.8.11 1023 30203 = 1.5.5.8.11 1024 113147 = 5.5.5.8.11 1025 154619 = 7.5.5.8.11 1026 237563 = 11.5.5.8.11 1027 137339 = 6.7.5.8.11 1028 44171 = 2.1.6.8.11 1029 85643 = 4.1.6.8.11 1030 28619 = 1.4.6.8.11 1031 132299 = 6.4.6.8.11 1032 194507 = 9.4.6.8.11 1033 235979 = 11.4.6.8.11 1034 59723 = 2.10.6.8.11 1035 121931 = 5.10.6.8.11 1036 163403 = 7.10.6.8.11 1037 225611 = 10.10.6.8.11 1038 206603 = 9.11.6.8.11 1039 70379 = 3.4.8.8.11 1040 215531 = 10.4.8.8.11 1041 77291 = 3.8.8.8.11 1042 37547 = 1.9.8.8.11 1043 224171 = 10.9.8.8.11 1044 80747 = 3.10.8.8.11 1045 101483 = 4.10.8.8.11 1046 122219 = 5.10.8.8.11 1047 91811 = 4.5.1.6.11 1048 133283 = 6.5.1.6.11 1049 74531 = 3.7.1.6.11 1050 95267 = 4.7.1.6.11 1051 136739 = 6.7.1.6.11 1052 219683 = 10.7.1.6.11 1053 76259 = 3.8.1.6.11 1054 117731 = 5.8.1.6.11 1055 179939 = 8.8.1.6.11 1056 221411 = 10.8.1.6.11 1057 242147 = 11.8.1.6.11 1058 130259 = 6.3.4.6.11 1059 213203 = 10.3.4.6.11 1060 233939 = 11.3.4.6.11 1061 93971 = 4.6.4.6.11 1062 197651 = 9.6.4.6.11 1063 80147 = 3.10.4.6.11 1064 204563 = 9.10.4.6.11 1065 225299 = 10.10.4.6.11 1066 130547 = 6.3.6.6.11 1067 213491 = 10.3.6.6.11 1068 30323 = 1.5.6.6.11 1069 51059 = 2.5.6.6.11 1070 216947 = 10.5.6.6.11 1071 237683 = 11.5.6.6.11 1072 95987 = 4.7.6.6.11 1073 178931 = 8.7.6.6.11 1074 220403 = 10.7.6.6.11 1075 35507 = 1.8.6.6.11 1076 139187 = 6.8.6.6.11 1077 242867 = 11.8.6.6.11 1078 65027 = 3.1.7.6.11 1079 87491 = 4.2.7.6.11 1080 232643 = 11.2.7.6.11 1081 33923 = 1.7.7.6.11 1082 116867 = 5.7.7.6.11 1083 199811 = 9.7.7.6.11 1084 39107 = 1.10.7.6.11 1085 122051 = 5.10.7.6.11 1086 142787 = 6.10.7.6.11 1087 184259 = 8.10.7.6.11 1088 148403 = 7.1.10.6.11 1089 231347 = 11.1.10.6.11 1090 113843 = 5.5.10.6.11 1091 176051 = 8.5.10.6.11 1092 158771 = 7.7.10.6.11 1093 101747 = 4.10.10.6.11 1094 205427 = 9.10.10.6.11 1095 246899 = 11.10.10.6.11 1096 34499 = 1.7.11.6.11 1097 117443 = 5.7.11.6.11 1098 138179 = 6.7.11.6.11 1099 179651 = 8.7.11.6.11 1100 101891 = 4.10.11.6.11 1101 226307 = 10.10.11.6.11 1102 105767 = 5.1.2.5.11 1103 188711 = 9.1.2.5.11 1104 91943 = 4.5.2.5.11 1105 55799 = 2.8.3.5.11 1106 225143 = 10.10.3.5.11 1107 49031 = 2.4.4.5.11 1108 69767 = 3.4.4.5.11 1109 33479 = 1.7.4.5.11 1110 116423 = 5.7.4.5.11 1111 87623 = 4.2.8.5.11 1112 108359 = 5.2.8.5.11 1113 212039 = 10.2.8.5.11 1114 39239 = 1.10.8.5.11 1115 246599 = 11.10.8.5.11 1116 61703 = 2.11.8.5.11 1117 123911 = 5.11.8.5.11 1118 165383 = 7.11.8.5.11 1119 186119 = 8.11.8.5.11 1120 48023 = 2.3.9.5.11 1121 151703 = 7.3.9.5.11 1122 172439 = 8.3.9.5.11 1123 51479 = 2.5.9.5.11 1124 92951 = 4.5.9.5.11
Comparisons
The Fortran version checks for possible factors up to 2800 by bignumber division before it starts on the MR process. Dinosaur (talk) 06:11, 9 April 2017 (UTC)
- One of the purposes of this task is to highlight problems with primality testing of a large number of large numbers. If you are using MR these are twofold: first being able to randomly select prime numbers fairly across the range 3 to the large number; second the number of prime numbers you are going to try. In the worst case for a single trial there is a 1 in 4 (usual case more like 1 in 50) chance that the large number is actually composite. This chance rapidly decreases with trials, but so does the time taken. So the recommendation is to do quick trials allowing some non prime candidates to be propagated, then a thorough check of your final candidate(s) that it is indeed a left truncatable prime.--Nigel Galloway (talk) 11:55, 9 April 2017 (UTC)
- Indeed. An earlier version accidentally used a parameter BIGMRTRIALS instead of what I intended for the upper bound of direct division trials, 2800 (eventually chosen after some ad-hoc timing as resulting in faster runs than with say 3000 or 1500), and with a value of six, this meant that only 2, 3, and 5 were tested as possible factors by direct division. All else was via the MR test and so far, this provided the only occasion when a denunciation of "composite" did not occur in the first MR trial, though only for a few dozen out of a quarter of a million trials. With trial factors up to 2800 or so, all denunciations were made on the first MR trial: 517641 trials rejecting 282261 in the first go, and accepting 235380 in all of the second to sixth trials. This suggests that "no comment" from a MR trial might be more likely for numbers having a small prime factor, however, I'm sure this question has been investigated more thoroughly than that. So, in this case, as the horde of candidates is being produced then trimmed one MR trial would have sufficed, but this is hindsight.
- Allowing a few extra (but wrong) candidates during the marshalling by avoiding repeated MR tests would save time that could be spent on more heavily testing the final few survivors, however, remember that the process is to produce not just a final number that is prime, but one that also is prime at each of its earlier stages. In other words, suppose a non-prime is admitted at say the five-digit level, and at the end (with say seven digits), some of its descendants may survive and turn out to be primes and possibly even the largest survivor. But alas, not all of its left-truncations (to six, five, four, etc. digits) would be primes.
- Routine BIGMRPRIME does not try to be general and produce tests using the range [2:N - 1] for a bignumber N because of the nuisances of bignumber arithmetic when the number is big, just an ad-hoc trick with an upper bound of N1 = min(N - 1,20000000) then selecting A1 via pseudo-random function RAND into the range [2,N1] - with vexation since RAND can deliver 1 as its result. In other words, the full range of A1 in [2,N - 1] is not available for numbers above twenty million, but they seemed to work well enough that I didn't bother with yet more routines for bignumber arithmetic to handle this properly. Incidentally, the Miller–Rabin_primality_test page quotes two as the lower bound for A1, not three. A value of two in a binary computer could enable shortcuts in the calculations, for a faster initial trial before continuing with the full range of possibilities for A1. Somewhere I saw some remarks that certain trial values of A1 give definite results for N up to ... except that I didn't look closely. There might even be a collection of special values that are particularly effective at denouncing composite numbers, rather than floundering about with "random" choices... Dinosaur (talk) 12:02, 10 April 2017 (UTC)
- One of the purposes of this task is to highlight problems with primality testing of a large number of large numbers. If you are using MR these are twofold: first being able to randomly select prime numbers fairly across the range 3 to the large number; second the number of prime numbers you are going to try. In the worst case for a single trial there is a 1 in 4 (usual case more like 1 in 50) chance that the large number is actually composite. This chance rapidly decreases with trials, but so does the time taken. So the recommendation is to do quick trials allowing some non prime candidates to be propagated, then a thorough check of your final candidate(s) that it is indeed a left truncatable prime.--Nigel Galloway (talk) 11:55, 9 April 2017 (UTC)
Odd base optimization
Sadly 24 is not odd, but for bases which are:
- Note that all stems greater than 2 are odd.
- Prepending an odd number to the stem makes a new even candidate stem, which can not be prime.
Therefore it is only necessary to consider cases where the number prepended to the stem is even.
For 2 the situation is the reverese. --Nigel Galloway 12:27, 10 December 2012 (UTC)