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 p^10 is special (Posted on 2015-06-12)
What prime number, raised to the tenth power, creates a number containing all ten digits?

I know the smallest one, you are free to list few followers.

 No Solution Yet Submitted by Ady TZIDON No Rating

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 computer solution Comment 1 of 1
As the lowest p^10 exceeds the precision available in Visual Basic (it has 19 digits, as seen below), UBASIC was used, under DOSBox:

5   open "p10spcl.txt" for output as #2
10     while P10<99999999999999999999999999999999
20        P=nxtprm(P)
30        P10=P*P*P*P*P:P10=P10*P10
40        S\$=str(P10)
50        Srce\$="0123456789"
60        Good=1
70        for I=1 to 10
80          if instr(S\$,mid(Srce\$,I,1))=0 then Good=0:cancel for:goto 100
90        next
100        if Good then
105            :if Ct>0 then print Ct:print #2,Ct:endif
110            :print P;str(P10):print #2,P;str(P10):Ct=0
120         :else inc Ct
130     wend
150   close #2

The 19th prime, 67, is the lowest whose 10th power contains all 10 digits.

The numbers intervening between the primes and their 10th powers that fit the description listed below show the number of intervening primes that fail to meet the criterion.  As the 10th powers become larger, each one has a greater likelihood of having all the digits, so the gaps between non-fits become smaller and less frequent.

18
67  1822837804551761449
3
83  15516041187205853449
9
137  2329194047563391944849
2
151  6162677950336718514001
1
163  13239635967018160063849
1
173  24013807852610947920649
6
211  174913992535407978606601
1
227  363294289954110647868649
9
277  2659485890900719634874649
1
283  3295067800663118480459449
293  4663115832631376345704249
307  7436759805837217107346249
311  8464550303319308288547601
2
331  15786284949774657045043801
337  18892916655137732057698849
347  25310151684662980796181049
349  26807373765254438673009001
3
373  52130071199257068815346649
3
397  97253461433805715000527049
3
421  174912544792453358346502201
1
433  231674888957051615288276449
439  265855226381706476903427601
2
457  397339737654378065640319249
461  433520115685490720702646601
463  452699390921229872008282849
1
479  635850671321437457459323201
487  750394448018639455972876849
491  814357163924275931995589401
499  957206097023388110011245001
503  1036763861454476003909724049
1
521  1473591924952786487925133201
523  1531147003165845604229778649
541  2147695222527137498891207401
6
587  4857144371993096171902453849
2
601  6148153756249382761936206001
1
613  7492120352012921267174929849
2
631  10006762501329395540915206801
641  11710630960672484548954118401
2
653  14097165966879236484266511049
659  15447355662831141299710475401
2
677  20224914151912685933807022649
683  22090575837180674640752471449
691  24818782535914467768701411401
701  28653665037714694102182057001
709  32096845506516383920668257401
719  36922313359187619548244760801
2
739  48578531695030075776442554601
4
769  72320829380154899085462412801
3
809  120084019435520610159775496401
811  123085967045938218547531752601
1
823  142560218431877459191575717649
827  149642548378589035457984790649
829  153301112139724882769277052201
1
853  203934316504189676181259377049
857  213701819195980161082078147249
6
907  376767011703542749261101388249
1
919  429689868625634225501910718801
2
941  544372570681789282481196251401
947  580095892065127629623589183049
953  617915381969049828620453160049
1
971  745061803065730690750719610201
977  792402135334027660597541583649
1
991  913558883040682586951726894401
1
1009  1093733872802526507260136674401
1013  1137874732397032526105297221849
1019  1207096081374615112024889047801
1
1031  1357021263671984015854738690801
1
1039  1466072594754994452219317781601
1
1051  1644474563565989751942777123001
1
1063  1842182469756309773938865020849
1069  1948843891164479242385262049801
1
1091  2389172492422689937648139855401
1
1097  2523865899030501209758389996049
1103  2665355388374552289094103022049
1109  2813944072277932332748750613401
1117  3023651481566388603275857817449
1123  3190050532444758047838177756649
1129  3364646318655149518033174309201
1
1153  4152341586657848098473435426049
1
1171  4848070768657369381097155672201
1181  5278359623878552774340709709801
1187  5552738093466546144133466815849
1193  5839886905510184257662663991249
1
1213  6896170790228275534651691767849
1217  7126983975202027668329968664449
1
1229  7861742750443423214189056528201
1231  7990620944964601562553351832801
1237  8388745064171653982358874141849
1249  9238987589682764184336007800001
1259  10005926331657404826061000709401
1
1279  11714006071361516311685439475201
1283  12085554015438371186846267489449
1289  12662783263238858255384077491601
1291  12860635297577242291503992477401
1
1301  13892262011451220148404716063001
1
1307  14546411725825779139203515416249
2
1327  16932037772659338838775736975649
1361  21806263317960671515439166765601
1367  22786894713095597130003654122449
1373  23807038395436052116504126476649
1381  25231134911761774155441593551801
1399  28719517871453000049926808186001
1409  30839684831314582312276485830401
3
1433  36514209582527818811780028806449
1
1447  40242533215980874846883634768049
1451  41368914405341012492749742127001
1453  41942676713311852147148995575049
1
1471  47438038942261552120809896515201
1
1483  51453095802482700024198747695449
1
1489  53573124675257581527363423309601
1493  55029818098573228516291079620249
2
1523  67142735848150219470136134408649
1531  70754148210526575834581598295801
1543  76499616963756636870228467179249
1
1553  81604584391921475272607698758049
1559  84812751313490671426290625976401
1
1571  91571820943889124079441046596201
1579  96343251677561117001381202882201
1
1597  107906886404720396377065272331049

 Posted by Charlie on 2015-06-12 11:01:57

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