> 8: 6666606077777070555550509999909011111010 The ninth term for the 19-divisibility criterion is 999999982. I'm trying for a tenth term but it appears to be a lengthy compute. I've submitted seven or eight terms for every n-divisibility criterion for n=11 to n=99 to the Integer Sequence database and they are numbered A093211-A093299: Last two digits = n. Neil has also posted my tabular form for these sequences (term #2 - term #7 only) here: http://www.research.att.com/~njas/sequences/a093211.txt I'll likely be updating this to term #9 or term #10 (if the compute times don't prove too exorbitant) in the future. Hans Havermann > ... if I'm doing this right, the length-5 substrings maximum is > 1100133003990098800855005 And my candidates for length-6 through length-8 maximums are: 6: 99960944490466640699960 7: 444440433333037777707 8: 6666606077777070555550509999909011111010 I think we can generate a sequence like this for almost any number. Here's 23: 1: 0 2: 46923 3: 5759897 4: 340483036808 5: 5554533373 6: 999994888881222226 7: 220202055050501101010 8: 11111001222220024444400488888008 Hans Havermann Hi Ed, I get 77714446555. The repeating-digit thing (after 722855133 for length-3 substrings) might indeed suggest a pattern but, if I'm doing this right, the length-5 substrings maximum is 1100133003990098800855005. Hans Havermann ----------------------------------------------------------------------- Good afternoon, The largest number so that all length 2 substrings are distinct and divisible by 19 is 19576 As far as the length 4 substrings are concerned, hereafter the results that I have got with the divisor d=19 ,then with the prime numbers from 13 to 97 when there exist more than one number. d root following figures 19 7771 4 4 4 6 5 5 5 13 1001 3 0 0 3 9 0 0 9 1 0 0 1 17 9996 2 2 2 7 8 8 8 23 3404 8 3 0 3 6 8 0 8 29 5452 4 9 4 31 7099 2 0 7 7 5 0 2 2 37 8880 6 6 6 0 41 9881 5 9 9 43 9202 1 5 0 5 47 8084 6 0 6 3 53 9063 6 0 4 2 4 0 67 8844 2 2 1 1 71 6603 5 5 0 79 7742 6 6 89 3382 7 7 97 1164 9 9 From Philippe Fondanaiche,Paris France ----------------------------------------------------------------------- I got 77714446555 for length 4 1100133003990098800855005 for length 5 - Brian Bitto ---------------------------------------------------------------------- These were actually easy to find with only a calculator, no need to be a "computer solver" (This isn't totally true - I used excell to make a large chart of mutiples of 19) a=length of substring (3 in your example) b=substring divisibility (19 in your example) F(a,b)= largest number such that every subtring of length a is divisible by b. F(2,19)=19576 F(3,19)=722855133 F(4,19)=77714446555 F(5,19)=1100133003990098800855005 F(6,19)=469646694969946494...(These 15 digits can repeat indefinitely) Some others F(2,17)=68517 F(3,17)=4935782 (also with 7 digits is 4595272) F(4,17)=99962227888 F(2,21)=8421 F(3,21)=252525... or 7777777... F(4,21)=77707770... Method: While trying to find F(3,19) I noticed that any time I used started with 3 digits beginning with a 1 I couldn't find a fourth digit, but if I began with 2, the fourth digit was always 5. So if a number, say 285 is divisible by 19, then 855 is too. 1000 = 12 (mod 19) 2000 = 5 (mod 19) this is the key. Removing the 2 and appending the 5 will keep the number divisible by 19. It doesn't work with 1 because you can't append a 12. 3000 = 17 no good 4000 = 10 no good 5000 = 3 GOOD 6000 = 15 no good 7000 = 8 GOOD 8000 = 1 GOOD 9000 = 13 no good 0 = 0 GOOD The third digit after a 2 must be a 5, etc... So the best numbers have 0, 2, 5, 7, and 8 in them. 0, 2 and 7 are best because they go to other good numbers. 722 is the only 3 digit number using these which is divisible by 19. Constructing gives 722855133 which stops here because 1 is no good. In general to find a solution: [1] Find the digits, d, for which d*10^a is a one digit number (mod b). [2] Determine which of these digits are best. [3] Find an a digit number divisible by b which uses only the best digits. [4] Try to construct the number The indefinite cases occurr where there is a loop of good numbers to other good numbers. For example F(6,19) arises from 9*10^6 = 4 (mod 19) 4*10^6 = 6 (mod 19) 6*10^6 = 9 (mod 19) So any 6 digit number consisting of only 0, 4, 6, and 9 which is divisible by 19 will yield an infinite loop. Fun -Jeremy Galvagni ----------------------------------------------------------------------- I think 7714446555 is the largest number so that all length 4 substrings are distinct and divisible by 19. cu Yves Hetzer ----------------------------------------------------------------------- 44393381886 - all 4 digit substrings are divisible by 23 regards Eswaran Narasimhan ----------------------------------------------------------------------- Dear Ed I could find 77714446555 for length 4 substrings and divisble by 19. regards Eswaran Narasimhan 2-d: 19576 3-d: 722855133 4-d: 77714446555 5-d: 1100133003990098800855005 6-d: infinite loop: 444999666444999666 etc. 446994669446994669 etc. 464949696464949696 etc. longer loops are possible: 449464664696996949449464664696996949 etc. Luke Pebody ----------------------------------------------------------------------- Hi Ed, > For hand solvers -- find the smallest number so that all length 2 substrings > are divisible by 19. For computer solvers, find the smallest number so that > all length 4 substrings are divisible by 19. I don't think you really want the smallest (that would be 19 and 1007) but the longest string, and then the smallest (non-zero) of that length. For n=19 I did a quick computer search for each substring length: 2:19576 3:722855133 4:77714446555 5:1100133003990098800855005 6:9999704444106666 7:444440433333037777707 8:6000606670007077500050559000909910001011 > Is there a pattern to this? I don't see it. An interesting case is substring length 6, where there are infinitely long (i.e. repeating) solutions, the 'smallest' of which is: 400064600096900049 400064600096900049 ...... The repeating block is of length 18, which is unsurprising since 19 divides 10^18-1 by Fermat's little theorem. > Can any number higher than 19 be used? I have tried a couple of other numbers, but none give strings as long as 19 does. The n=21 case is neat since it has repeating solutions for all substring lengths >2: 2:8421 3:[777] or [252] 4:[7077] 5:[70077] 6:[700077] or [100002] 7:[7000077] etc. --- I will also be at g4g6. It is my first time, and I'm really looking forward to it. Bye, Jaap ===== === Jaap Scherphuis Jaap's Puzzle Page: http://www.geocities.com/jaapsch/puzzles ----------------------------------------------------------------------- The longest string I could find was 77714446555. Jeff Smith ----------------------------------------------------------------------- Dear Ed, this is an email from Munich, Germany. Here are my findings of the problem of Erich Friedman - factors in substrings: In the enclosed files you find the solutions for the substring-lengths 2 till 7 with all possible factors. Results are only recorded when the total string is larger than the substring-length +1. And within the factors I only record increasingly better results, taking numbers with "00" only if they are longer. At the end of each file you find the maximum result for this substring-length. The bigger the factor is, the less likely it is to get longer solutions. So the smallest factor "11" has most solutions. There are some nice solutions like: substring=4, factor=14, solution=991288484424228288 substring=4, factor=17, solution=99962227888 substring=4, factor=19, solution=77714446555 substring=5, factor=13, solution=5505577077220228808866066 substring=5, factor=17, solution=9990933303111016660622202 substring=5, factor=19, solution=1100133003990098800855005 substring=5, factor=813, solution=951219512 substring=5, factor=1061, solution=8488063660 substring=5, factor=2459, solution=6639344262 substring=6, factor=17, solution=800088400044200022100011900099 substring=6, factor=31, solution=111011222022444044888088 substring=6, factor=34, solution=888080444040222020 substring=6, factor=41, solution=555550999990888880 substring=6, factor=127, solution=211201422402844804 substring=6, factor=1002, solution=111222444888 substring=6, factor=15873, solution=98412698412 substring=6, factor=47619, solution=95238095238 substring=6, factor=142857, solution=14285714285 substring=7, factor=17, solution=70007701000110500055080008806000660 substring=7, factor=81841, solution=5355648535564 If there is a big prime-factor in the factor than it is more likely that the solution is bigger and those prime-factors which have a long decimal period for 1/prime. I hope, that you like some of my findings. Yours - Bodo (Bodo Zinser, Munich) Length: 2 12: 12484 72484 13: 1391 2652 3913 5265 6526 9139 14: 14284 98428 16: 164800 17: 68517 19: 19576 21: 8421 23: 46923 24: 7248 25: 2500 Max: 2 / 16: 164800 Length: 3 11: 20990220 22099022 24297924 25396825 26495726 27594627 28693528 29792429 30880330 11: 33088033 35286935 36385836 37484737 38583638 39682539 40770440 42979242 44077044 11: 46275946 47374847 48473748 49572649 50660550 52869352 53968253 55066055 57264957 11: 58363858 59462759 60550660 62759462 63858363 64957264 66055066 68253968 69352869 11: 70440770 72649572 73748473 74847374 75946275 77044077 79242979 80330880 82539682 11: 83638583 84737484 85836385 86935286 88033088 90220990 92429792 93528693 94627594 11: 95726495 96825396 97924297 99022099 12: 108408 1204804 13240840 15648048 19240840 21684084 27684084 31204804 37204804 12: 43240840 45648048 49240840 51684084 57684084 61204804 67204804 73240840 75648048 12: 79240840 81684084 87684084 91204804 97204804 13: 40390 44299 45598845 46897546 50780650 54689754 55988455 59884559 65078065 68975468 13: 75468975 78065078 80650780 84559884 88455988 89754689 97546897 98845598 14: 11266 15462 16868 18266 308406 336448 350420 378406 518266 560280 588266 616868 14: 6308406 6588266 7560280 15: 22555 28555 52555 58555 82555 88555 16: 11288000 17: 30680 340850 4595272 4935782 18: 28828 46846 48648 64864 68468 82882 84684 86486 88288 19: 20950 22855133 722855133 20: 12000 21: 27357 50420 75672 22: 30880 330880 550660 770440 990220 23: 50690 55299 5759897 24: 26480 336000 552888 25: 125000 26: 57286 59884 70260 75468 27: 10810 13513 16216 18918 21621 24324 27027 29729 32432 35135 37837 40540 43243 27: 45945 48648 51351 54054 56756 59459 62162 64864 67567 70270 72972 75675 78378 27: 81081 83783 86486 89189 91891 94594 97297 28: 30840 336448 30: 33000 31: 1558992 32: 19288 41608 448000 5448000 34: 30680 35: 77000 37: 14814 18518 25925 29629 37037 40740 48148 51851 59259 62962 70370 74074 81481 37: 85185 92592 96296 39: 50780 40: 12000 42: 50420 44: 30880 70440 45: 99000 50: 15000 54: 48648 64864 86486 55: 60550 660550 56: 67280 57: 22855 60: 36000 62: 18682 80620 70: 77000 80: 24000 90: 99000 120: 36000 125: 125000 160: 48000 250: 25000 Max: 3 / 19: 722855133 Length: 4 11: 1001100 1012101 1023102 1034103 1045104 1056105 1067106 1078107 1089108 1122112 11: 1133113 1144114 1155115 1166116 1177117 1188118 1199119 1210121 1221122 1232123 11: 1243124 1254125 1265126 1276127 1287128 1298129 1309130 1320132 1331133 1342134 11: 1353135 1364136 1375137 1386138 1397139 1408140 1419141 1430143 1441144 1452145 11: 1463146 1474147 1485148 1496149 1507150 1518151 1529152 1540154 1551155 1562156 11: 1573157 1584158 1595159 1606160 1617161 1628162 1639163 1650165 1661166 1672167 11: 1683168 1694169 1705170 1716171 1727172 1738173 1749174 1760176 1771177 1782178 11: 1793179 1804180 1815181 1826182 1837183 1848184 1859185 1870187 1881188 1892189 11: 1903190 1914191 1925192 1936193 1947194 1958195 1969196 1980198 1991199 2013201 11: 2024202 2035203 2046204 2057205 2068206 2079207 2090209 2101210 2112211 2123212 12: 10084008 1020408040 10324008400 10564080408 11164440444 11524488448 11764440444 12: 13564080408 14164440444 14524488448 14764440444 16564080408 17164440444 17524488448 12: 17764440444 19564080408 20168040804 20528088808 20768040804 21128448844 21728448844 12: 22328808880 22568880888 22928808880 23168040804 23528088808 23768040804 24128448844 12: 24728448844 25328808880 25568880888 25928808880 26168040804 26528088808 26768040804 12: 27128448844 27728448844 28328808880 28568880888 28928808880 29168040804 29528088808 12: 29768040804 31320408040 31920408040 32520888088 34320408040 34920408040 35520888088 12: 37320408040 37920408040 38520888088 40564080408 41164440444 41524488448 41764440444 12: 43564080408 44164440444 44524488448 44764440444 46564080408 47164440444 47524488448 12: 47764440444 49564080408 50168040804 50528088808 50768040804 51128448844 51728448844 12: 52328808880 52568880888 52928808880 53168040804 53528088808 53768040804 54128448844 12: 54728448844 55328808880 55568880888 55928808880 56168040804 56528088808 56768040804 12: 57128448844 57728448844 58328808880 58568880888 58928808880 59168040804 59528088808 12: 59768040804 61320408040 61920408040 62520888088 64320408040 64920408040 65520888088 12: 67320408040 67920408040 13: 100130039009100 105330299061105 106630559022106 109230169035109 110533029906110 13: 113133939919113 119633159932119 120936019503120 122236669555122 123536929516123 13: 126136539529126 131339399191131 132639659152132 133939919113133 135239269165135 13: 136539529126136 152132639659152 156032509620156 159932119633159 161235369295161 13: 162535629256162 165135239269165 169035109230169 191131339399191 195031209360195 13: 196331599321196 201560325096201 209360195031209 210663055902210 211963315993211 13: 213263965915213 221066305590221 222366695551222 223666955512223 226266565525226 13: 230169035109230 235369295161235 236669555122236 239269165135239 250962015603250 13: 252262665655252 253562925616253 256162535629256 261365395291261 262665655252262 13: 263965915213263 266565525226266 269165135239269 291261365395291 292561625356292 13: 295161235369295 299061105330299 301690351092301 302990611053302 305590221066305 13: 312093601950312 313393991911313 315993211963315 321196331599321 325096201560325 13: 326396591521326 330299061105330 331599321196331 339399191131339 351092301690351 13: 352392691651352 353692951612353 356292561625356 360195031209360 365395291261365 13: 366695551222366 369295161235369 392691651352392 393991911313393 395291261365395 13: 396591521326396 399191131339399 503120936019503 509620156032509 510923016903510 13: 512223666955512 513523926916513 516123536929516 521326396591521 522626656552522 13: 523926916513523 525226266565525 529126136539529 533029906110533 535629256162535 13: 536929516123536 539529126136539 551222366695551 552522626656552 555122236669555 13: 559022106630559 560325096201560 561625356292561 562925616253562 565525226266565 14: 1008400420028008 1022408820448022 107840042002800840 109240882044802240 119044802240882044 14: 121848442422828848 127448022408820448 141442422828848442 179240882044802240 189044802240882044 14: 191848442422828848 197448022408820448 211484424228288484 219884844242282884 221288484424228288 14: 247882044802240882 249282884844242282 281484424228288484 289884844242282884 291288484424228288 14: 401820448022408820 417224088204480224 429828848442422828 441022408820448022 449422828848442422 14: 471820448022408820 487224088204480224 499828848442422828 711204480224088204 729408820448022408 14: 781204480224088204 799408820448022408 809240882044802240 819044802240882044 821848442422828848 14: 827448022408820448 841442422828848442 879240882044802240 889044802240882044 891848442422828848 14: 897448022408820448 911484424228288484 919884844242282884 921288484424228288 947882044802240882 14: 949282884844242282 981484424228288484 989884844242282884 991288484424228288 15: 201050 2025505550 2085505550 2220555055 2235550555 2265550555 2280555055 2295550555 15: 2325505550 2385505550 2520555055 2535550555 2565550555 2580555055 2595550555 15: 2625505550 2685505550 2820555055 2835550555 2865550555 2880555055 2895550555 15: 2925505550 2985505550 3225055505 3285055505 3525055505 3585055505 3825055505 15: 3885055505 5025505550 5085505550 5220555055 5235550555 5265550555 5280555055 15: 5295550555 5325505550 5385505550 5520555055 5535550555 5565550555 5580555055 15: 5595550555 5625505550 5685505550 5820555055 5835550555 5865550555 5880555055 15: 5895550555 5925505550 5985505550 6225055505 6285055505 6525055505 6585055505 15: 6825055505 6885055505 8025505550 8085505550 8220555055 8235550555 8265550555 15: 8280555055 8295550555 8325505550 8385505550 8520555055 8535550555 8565550555 15: 8580555055 8595550555 8625505550 8685505550 8820555055 8835550555 8865550555 15: 8880555055 8895550555 8925505550 8985505550 9225055505 9285055505 9525055505 15: 9585055505 9825055505 9885055505 16: 10080000 17: 1003400 10204080 11054403 11224488 11564437 11904420 12924828 16154743 16664777 17: 19214284 19554233 20918024 21258483 21598432 22108840 22618874 22958823 25168347 17: 25508330 26018704 26528738 26698772 29928228 50153043 50663077 51513434 52023808 17: 52193842 55253383 55593332 56103740 56613774 56953723 59163247 59503230 60527038 17: 60697072 61207480 62057803 62227888 62567837 62907820 65117344 65627378 65967327 17: 69027208 69197242 9010204080 9061207480 90952023808 99962227888 18: 201620 203420 205220 207020 2088208 2268226 2286228 2448244 2466246 2484248 2628262 18: 2646264 2664266 2682268 2808280 2826282 2844284 2862286 2880288 2898280828 4698460846 18: 4896480648 6498640864 6696660666 6894680468 8298820882 8496840684 8694860486 18: 8892880288 19: 10076004 11406650 14446555 17106460 40475054 41045605 7011406650 7714446555 77714446555 20: 1020000 21: 1008400 1029408 10714074 11764473 12604830 16174347 17014704 17224788 20168043 21: 21218484 22268883 26678337 27728778 60273087 61113444 62163843 67203780 67623738 21: 716174347 726678337 760273087 22: 2002200 2024202 2046204 2068206 2244224 2266226 2288228 2420242 2442244 2464246 22: 2486248 2640264 2662266 2684268 2860286 2882288 4026402 4048404 4202420 4224422 22: 4246424 4268426 4422442 4466446 4488448 4620462 4642464 4664466 4686468 4840484 22: 4862486 4884488 6028602 6204620 6226622 6248624 6402640 6424642 6446644 6468646 22: 6622662 6644664 6688668 6820682 6842684 6864686 6886688 8206820 8228822 8404840 22: 8426842 8448844 8602860 8624862 8646864 8668866 8822882 8844884 8866886 23: 301380 3036808 3082806 3335888 3381886 340483036808 24: 201680 204080 2064800 20888088 220888088 22328808880 22568880888 23528088808 24: 25928808880 28328808880 28568880888 29528088808 35520888088 52328808880 52568880888 24: 53528088808 55928808880 58328808880 58568880888 59528088808 65520888088 82328808880 24: 82568880888 83528088808 85928808880 88328808880 88568880888 89528088808 95520888088 25: 10250000 26: 20026006 25226266 5070208060 27: 302430 305130 307830 332133 334833 337533 361836 364536 367236 3699369 3969396 27: 3996399 6399639 6669666 6696669 6939693 6966696 6993699 9099909 9369936 9396939 27: 9639963 9666966 9693969 9909990 9936993 9963996 9990999 28: 10084004 10924088 11204480 12884844 17924088 18204480 19884844 207280084004 29: 4002900 4408990 4553944 5046409 5452494 30: 3030000 31: 201550 204650 20775022 27905270 709920775022 32: 201600 20480000 33: 1023102 1056105 1089108 1122112 1155115 1188118 1221122 1254125 1287128 1320132 33: 1353135 1386138 1419141 1452145 1485148 1518151 1551155 1584158 1617161 1650165 33: 1683168 1716171 1749174 1782178 1815181 1848184 1881188 1914191 1947194 1980198 33: 2013201 2046204 2079207 2112211 2145214 2178217 2211221 2244224 2277227 2310231 33: 2343234 2376237 2409240 2442244 2475247 2508250 2541254 2574257 2607260 2640264 33: 2673267 2706270 2739273 2772277 2805280 2838283 2871287 2904290 2937293 2970297 33: 3036303 3069306 3102310 3135313 3168316 3201320 3234323 3267326 3366336 3399339 33: 3432343 3465346 3498349 3531353 3564356 3597359 3630363 3663366 3696369 3729372 33: 3762376 3795379 3828382 3861386 3894389 3927392 3960396 3993399 4026402 4059405 33: 4092409 4125412 4158415 4191419 4224422 4257425 4290429 4323432 4356435 4389438 34: 10204080 11224488 11904420 12924828 22108840 29928228 9010204080 35: 301050 304550 308050 332555 336055 339555 371050 374550 378050 7035005 36: 403240 406840 442844 446444 482448 486048 4896480 8496840 8892880 37: 403330 407030 4403330 44403330 48843663 88806660 38: 102660 106460 1102660 1178664 17106460 39: 30039009 40: 1040000 41: 803690 807790 881599 885699 8897995 9881599 42: 1008400 1218484 2016804 21008400 43: 2021505 2924515 92021505 44: 4004400 4048404 4488448 4840484 4884488 8404840 8448844 8844884 45: 504050 508550 553555 558055 594050 598550 9090000 46: 3036808 3082806 8832668 47: 4042303 4089306 4841363 48883666 80846063 48: 302400 3360000 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