Multiply Pandigital 2

davar55 · 2013-01-07, 02:24

What is the smallest positive integer n such that
both 2^n and 3^n are each singly pandigital, i.e.
contain all ten digits at least once? What about
doubly pandigital, i.e. contain each of all ten digits
at least twice? Care to try for three?

(Based on the "Multiply Pandigital" thread.)

LaurV · 2013-01-07, 07:23

You imagine people have nothing to do and waste their time with trifles?

Code:

(14:08:12) gp > pandig(n)=v=[0,0,0,0,0,0,0,0,0,0];while(n>0,v[n%10+1]++;n\=10);return(vecsort(v)[1])
%4 = (n)->v=[0,0,0,0,0,0,0,0,0,0];while(n>0,v[n%10+1]++;n\=10);return(vecsort(v)[1])
(14:09:32) gp > n=15; until(b>0&&d>0,print(n++", "a=2^n", "b=pandig(a)"   :   "c=3^n", "d=pandig(c)))
16, 65536, 0   :   43046721, 0
17, 131072, 0   :   129140163, 0
18, 262144, 0   :   387420489, 0
19, 524288, 0   :   1162261467, 0
20, 1048576, 0   :   3486784401, 0
21, 2097152, 0   :   10460353203, 0
22, 4194304, 0   :   31381059609, 0
23, 8388608, 0   :   94143178827, 0
24, 16777216, 0   :   282429536481, 0
25, 33554432, 0   :   847288609443, 0
26, 67108864, 0   :   2541865828329, 0
27, 134217728, 0   :   7625597484987, 0
28, 268435456, 0   :   22876792454961, 0
29, 536870912, 0   :   68630377364883, 0
30, 1073741824, 0   :   205891132094649, 0
31, 2147483648, 0   :   617673396283947, 0
32, 4294967296, 0   :   1853020188851841, 0
33, 8589934592, 0   :   5559060566555523, 0
34, 17179869184, 0   :   16677181699666569, 0
35, 34359738368, 0   :   50031545098999707, 0
36, 68719476736, 0   :   150094635296999121, 0
37, 137438953472, 0   :   450283905890997363, 0
38, 274877906944, 0   :   1350851717672992089, 0
39, 549755813888, 0   :   4052555153018976267, 1
40, 1099511627776, 0   :   12157665459056928801, 0
41, 2199023255552, 0   :   36472996377170786403, 0
42, 4398046511104, 0   :   109418989131512359209, 0
43, 8796093022208, 0   :   328256967394537077627, 0
44, 17592186044416, 0   :   984770902183611232881, 0
45, 35184372088832, 0   :   2954312706550833698643, 1
46, 70368744177664, 0   :   8862938119652501095929, 0
47, 140737488355328, 0   :   26588814358957503287787, 1
48, 281474976710656, 0   :   79766443076872509863361, 1
49, 562949953421312, 0   :   239299329230617529590083, 0
50, 1125899906842624, 0   :   717897987691852588770249, 0
51, 2251799813685248, 0   :   2153693963075557766310747, 0
52, 4503599627370496, 0   :   6461081889226673298932241, 0
53, 9007199254740992, 0   :   19383245667680019896796723, 1
54, 18014398509481984, 0   :   58149737003040059690390169, 0
55, 36028797018963968, 0   :   174449211009120179071170507, 0
56, 72057594037927936, 0   :   523347633027360537213511521, 0
57, 144115188075855872, 0   :   1570042899082081611640534563, 1
58, 288230376151711744, 0   :   4710128697246244834921603689, 0
59, 576460752303423488, 0   :   14130386091738734504764811067, 0
60, 1152921504606846976, 0   :   42391158275216203514294433201, 1
61, 2305843009213693952, 0   :   127173474825648610542883299603, 2
62, 4611686018427387904, 0   :   381520424476945831628649898809, 1
63, 9223372036854775808, 0   :   1144561273430837494885949696427, 1
64, 18446744073709551616, 0   :   3433683820292512484657849089281, 1
65, 36893488147419103232, 0   :   10301051460877537453973547267843, 1
66, 73786976294838206464, 0   :   30903154382632612361920641803529, 0
67, 147573952589676412928, 0   :   92709463147897837085761925410587, 2
68, 295147905179352825856, 1   :   278128389443693511257285776231761, 0
69, 590295810358705651712, 0   :   834385168331080533771857328695283, 1
70, 1180591620717411303424, 1   :   2503155504993241601315571986085849, 1

(14:10:07) gp > n=70; until(b>1&&d>1,print(n++", "a=2^n", "b=pandig(a)"   :   "c=3^n", "d=pandig(c)))
71, 2361183241434822606848, 0   :   7509466514979724803946715958257547, 1
72, 4722366482869645213696, 0   :   22528399544939174411840147874772641, 1
73, 9444732965739290427392, 0   :   67585198634817523235520443624317923, 1
74, 18889465931478580854784, 0   :   202755595904452569706561330872953769, 1
76, 75557863725914323419136, 0   :   1824800363140073127359051977856583921, 2
77, 151115727451828646838272, 0   :   5474401089420219382077155933569751763, 2
78, 302231454903657293676544, 0   :   16423203268260658146231467800709255289, 2
79, 604462909807314587353088, 1   :   49269609804781974438694403402127765867, 1
80, 1208925819614629174706176, 0   :   147808829414345923316083210206383297601, 1
...
<snip: uninteresting (0,0) lines were skipped, due to size limit of the post>
...
82, 4835703278458516698824704, 1   :   1330279464729113309844748891857449678409, 1
83, 9671406556917033397649408, 0   :   3990838394187339929534246675572349035227, 1
84, 19342813113834066795298816, 1   :   11972515182562019788602740026717047105681, 0
85, 38685626227668133590597632, 0   :   35917545547686059365808220080151141317043, 2
86, 77371252455336267181195264, 0   :   107752636643058178097424660240453423951129, 2
87, 154742504910672534362390528, 1   :   323257909929174534292273980721360271853387, 1
88, 309485009821345068724781056, 2   :   969773729787523602876821942164080815560161, 2

(14:15:13) gp > n=88; until(b>2&&d>2,print(n++", "a=2^n", "b=pandig(a)"   :   "c=3^n", "d=pandig(c)))
89, 618970019642690137449562112, 1   :   2909321189362570808630465826492242446680483, 1
90, 1237940039285380274899124224, 0   :   8727963568087712425891397479476727340041449, 2
...
92, 4951760157141521099596496896, 0   :   78551672112789411833022577315290546060373041, 2
93, 9903520314283042199192993792, 0   :   235655016338368235499067731945871638181119123, 2
94, 19807040628566084398385987584, 1   :   706965049015104706497203195837614914543357369, 1
95, 39614081257132168796771975168, 1   :   2120895147045314119491609587512844743630072107, 2
96, 79228162514264337593543950336, 1   :   6362685441135942358474828762538534230890216321, 2
97, 158456325028528675187087900672, 1   :   19088056323407827075424486287615602692670648963, 2
98, 316912650057057350374175801344, 1   :   57264168970223481226273458862846808078011946889, 2
99, 633825300114114700748351602688, 0   :   171792506910670443678820376588540424234035840667, 2
100, 1267650600228229401496703205376, 1   :   515377520732011331036461129765621272702107522001, 0
101, 2535301200456458802993406410752, 1   :   1546132562196033993109383389296863818106322566003, 0
102, 5070602400912917605986812821504, 0   :   4638397686588101979328150167890591454318967698009, 1
103, 10141204801825835211973625643008, 1   :   13915193059764305937984450503671774362956903094027, 1
104, 20282409603651670423947251286016, 2   :   41745579179292917813953351511015323088870709282081, 0
105, 40564819207303340847894502572032, 1   :   125236737537878753441860054533045969266612127846243, 2
106, 81129638414606681695789005144064, 1   :   375710212613636260325580163599137907799836383538729, 0
107, 162259276829213363391578010288128, 0   :   1127130637840908780976740490797413723399509150616187, 2
108, 324518553658426726783156020576256, 0   :   3381391913522726342930221472392241170198527451848561, 2
109, 649037107316853453566312041152512, 1   :   10144175740568179028790664417176723510595582355545683, 3
110, 1298074214633706907132624082305024, 1   :   30432527221704537086371993251530170531786747066637049, 2
111, 2596148429267413814265248164610048, 1   :   91297581665113611259115979754590511595360241199911147, 1
112, 5192296858534827628530496329220096, 1   :   273892744995340833777347939263771534786080723599733441, 2
113, 10384593717069655257060992658440192, 2   :   821678234986022501332043817791314604358242170799200323, 2
114, 20769187434139310514121985316880384, 2   :   2465034704958067503996131453373943813074726512397600969, 2
115, 41538374868278621028243970633760768, 1   :   7395104114874202511988394360121831439224179537192802907, 1
116, 83076749736557242056487941267521536, 2   :   22185312344622607535965183080365494317672538611578408721, 2
117, 166153499473114484112975882535043072, 2   :   66555937033867822607895549241096482953017615834735226163, 4
118, 332306998946228968225951765070086144, 2   :   199667811101603467823686647723289448859052847504205678489, 3
119, 664613997892457936451903530140172288, 3   :   599003433304810403471059943169868346577158542512617035467, 2
120, 1329227995784915872903807060280344576, 2   :   1797010299914431210413179829509605039731475627537851106401, 2
121, 2658455991569831745807614120560689152, 1   :   5391030899743293631239539488528815119194426882613553319203, 1
122, 5316911983139663491615228241121378304, 1   :   16173092699229880893718618465586445357583280647840659957609, 4
123, 10633823966279326983230456482242756608, 1   :   48519278097689642681155855396759336072749841943521979872827, 2
124, 21267647932558653966460912964485513216, 1   :   145557834293068928043467566190278008218249525830565939618481, 3
125, 42535295865117307932921825928971026432, 2   :   436673502879206784130402698570834024654748577491697818855443, 3
126, 85070591730234615865843651857942052864, 2   :   1310020508637620352391208095712502073964245732475093456566329, 2
127, 170141183460469231731687303715884105728, 1   :   3930061525912861057173624287137506221892737197425280369698987, 2
128, 340282366920938463463374607431768211456, 1   :   11790184577738583171520872861412518665678211592275841109096961, 2
129, 680564733841876926926749214863536422912, 1   :   35370553733215749514562618584237555997034634776827523327290883, 3
130, 1361129467683753853853498429727072845824, 1   :   106111661199647248543687855752712667991103904330482569981872649, 4
131, 2722258935367507707706996859454145691648, 2   :   318334983598941745631063567258138003973311712991447709945617947, 2
132, 5444517870735015415413993718908291383296, 1   :   955004950796825236893190701774414011919935138974343129836853841, 3
133, 10889035741470030830827987437816582766592, 3   :   2865014852390475710679572105323242035759805416923029389510561523, 4
(14:15:34) gp >

Batalov · 2013-01-07, 07:47

He must have been right!

davar55 · 2013-01-07, 20:23

Certainly don't want to trifle with this quick (very nice btw) solution to
the 1-ply 2-ply 3-ply pandigital problem. I was originally going to ask
not for 1, 2, and 3-ply but for 100-, 200-, and 300-ply pandigitals, but
now I see the problems scale easily that far.

I might have said 2^n-1 and 3^n-2 and all prime, but that's going
a bit too far, don't you think?

2013-01-07, 02:24	#1
davar55 May 2004 New York City 2³×23² Posts	Multiply Pandigital 2 What is the smallest positive integer n such that both 2^n and 3^n are each singly pandigital, i.e. contain all ten digits at least once? What about doubly pandigital, i.e. contain each of all ten digits at least twice? Care to try for three? (Based on the "Multiply Pandigital" thread.)

2013-01-07, 20:23	#4
davar55 May 2004 New York City 2³·23² Posts	Certainly don't want to trifle with this quick (very nice btw) solution to the 1-ply 2-ply 3-ply pandigital problem. I was originally going to ask not for 1, 2, and 3-ply but for 100-, 200-, and 300-ply pandigitals, but now I see the problems scale easily that far. I might have said 2^n-1 and 3^n-2 and all prime, but that's going a bit too far, don't you think? Last fiddled with by davar55 on 2013-01-07 at 21:03

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Multiply By Drawing Lines	petrw1	Math	2	2014-05-20 06:13
Sequences using nine-digit pandigital numbers as start	ChristianB	Aliquot Sequences	16	2014-05-16 06:56
Multiply Pandigital	davar55	Puzzles	18	2010-12-22 22:07
Multiply	mgb	Lounge	0	2008-07-28 12:54
Critical bug in Gnu MP FFT-multiply code	ET_	Lounge	3	2004-03-11 16:24

2013-01-07, 07:47	#3
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2²·2,333 Posts	He must have been right!