Reaching for the Last Pandigital Factorial Number

This post is for a dear friend. Many years ago I shared with him a mathematical puzzle that we both enjoyed. Later I posted What’s the next base-ten non-pandigital factorial number after 41!? on Math Stack Exchange.

Here’s the gist of the problem.

A “factorial number” is any element of the image of the factorial function. A pandigital number (of a given base) is an integer whose digits include at least one of each digit. Naturally, a pandigital factorial number is a number which is both a factorial number and a pandigital number. Is there a largest pandigital factorial number? Or is it a countably-infinite set that can be put into a bijection with the naturals?

At the request of my friend I have posted a table with three columns. The first is simply non-negative integers 0-to-1000, labelled n. The second column last_200_is_pandigital is an indicator variable for whether the last 200 digits contain at least one of each digit. The third column, last_200, is the last 200 digits of the factorial number (regardless of whether the entire number is pandigital). Maybe this will reveal some interesting patterns.

The table is wide enough that scrolling left-to-right will be necessary… Sorry!

n	last_200_is_pandigital	last_200
0	False	1
1	False	1
2	False	2
3	False	6
4	False	24
5	False	120
6	False	720
7	False	5040
8	False	40320
9	False	362880
10	False	3628800
11	False	39916800
12	False	479001600
13	False	6227020800
14	False	87178291200
15	False	1307674368000
16	False	20922789888000
17	False	355687428096000
18	False	6402373705728000
19	False	121645100408832000
20	False	2432902008176640000
21	False	51090942171709440000
22	False	1124000727777607680000
23	True	25852016738884976640000
24	False	620448401733239439360000
25	False	15511210043330985984000000
26	False	403291461126605635584000000
27	True	10888869450418352160768000000
28	False	304888344611713860501504000000
29	False	8841761993739701954543616000000
30	False	265252859812191058636308480000000
31	True	8222838654177922817725562880000000
32	False	263130836933693530167218012160000000
33	True	8683317618811886495518194401280000000
34	True	295232799039604140847618609643520000000
35	True	10333147966386144929666651337523200000000
36	True	371993326789901217467999448150835200000000
37	True	13763753091226345046315979581580902400000000
38	False	523022617466601111760007224100074291200000000
39	True	20397882081197443358640281739902897356800000000
40	True	815915283247897734345611269596115894272000000000
41	False	33452526613163807108170062053440751665152000000000
42	True	1405006117752879898543142606244511569936384000000000
43	True	60415263063373835637355132068513997507264512000000000
44	True	2658271574788448768043625811014615890319638528000000000
45	True	119622220865480194561963161495657715064383733760000000000
46	True	5502622159812088949850305428800254892961651752960000000000
47	True	258623241511168180642964355153611979969197632389120000000000
48	True	12413915592536072670862289047373375038521486354677760000000000
49	True	608281864034267560872252163321295376887552831379210240000000000
50	True	30414093201713378043612608166064768844377641568960512000000000000
51	True	1551118753287382280224243016469303211063259720016986112000000000000
52	True	80658175170943878571660636856403766975289505440883277824000000000000
53	True	4274883284060025564298013753389399649690343788366813724672000000000000
54	True	230843697339241380472092742683027581083278564571807941132288000000000000
55	True	12696403353658275925965100847566516959580321051449436762275840000000000000
56	True	710998587804863451854045647463724949736497978881168458687447040000000000000
57	True	40526919504877216755680601905432322134980384796226602145184481280000000000000
58	True	2350561331282878571829474910515074683828862318181142924420699914240000000000000
59	True	138683118545689835737939019720389406345902876772687432540821294940160000000000000
60	True	8320987112741390144276341183223364380754172606361245952449277696409600000000000000
61	True	507580213877224798800856812176625227226004528988036003099405939480985600000000000000
62	True	31469973260387937525653122354950764088012280797258232192163168247821107200000000000000
63	True	1982608315404440064116146708361898137544773690227268628106279599612729753600000000000000
64	True	126886932185884164103433389335161480802865516174545192198801894375214704230400000000000000
65	True	8247650592082470666723170306785496252186258551345437492922123134388955774976000000000000000
66	True	544344939077443064003729240247842752644293064388798874532860126869671081148416000000000000000
67	True	36471110918188685288249859096605464427167635314049524593701628500267962436943872000000000000000
68	True	2480035542436830599600990418569171581047399201355367672371710738018221445712183296000000000000000
69	True	171122452428141311372468338881272839092270544893520369393648040923257279754140647424000000000000000
70	True	11978571669969891796072783721689098736458938142546425857555362864628009582789845319680000000000000000
71	True	850478588567862317521167644239926010288584608120796235886430763388588680378079017697280000000000000000
72	True	61234458376886086861524070385274672740778091784697328983823014963978384987221689274204160000000000000000
73	True	4470115461512684340891257138125051110076800700282905015819080092370422104067183317016903680000000000000000
74	True	330788544151938641225953028221253782145683251820934971170611926835411235700971565459250872320000000000000000
75	True	24809140811395398091946477116594033660926243886570122837795894512655842677572867409443815424000000000000000000
76	True	1885494701666050254987932260861146558230394535379329335672487982961844043495537923117729972224000000000000000000
77	True	145183092028285869634070784086308284983740379224208358846781574688061991349156420080065207861248000000000000000000
78	True	11324281178206297831457521158732046228731749579488251990048962825668835325234200766245086213177344000000000000000000
79	True	894618213078297528685144171539831652069808216779571907213868063227837990693501860533361810841010176000000000000000000
80	True	71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000
81	True	5797126020747367985879734231578109105412357244731625958745865049716390179693892056256184534249745940480000000000000000000
82	True	475364333701284174842138206989404946643813294067993328617160934076743994734899148613007131808479167119360000000000000000000
83	True	39455239697206586511897471180120610571436503407643446275224357528369751562996629334879591940103770870906880000000000000000000
84	True	3314240134565353266999387579130131288000666286242049487118846032383059131291716864129885722968716753156177920000000000000000000
85	True	281710411438055027694947944226061159480056634330574206405101912752560026159795933451040286452340924018275123200000000000000000000
86	True	24227095383672732381765523203441259715284870552429381750838764496720162249742450276789464634901319465571660595200000000000000000000
87	True	2107757298379527717213600518699389595229783738061356212322972511214654115727593174080683423236414793504734471782400000000000000000000
88	True	185482642257398439114796845645546284380220968949399346684421580986889562184028199319100141244804501828416633516851200000000000000000000
89	True	16507955160908461081216919262453619309839666236496541854913520707833171034378509739399912570787600662729080382999756800000000000000000000
90	True	1485715964481761497309522733620825737885569961284688766942216863704985393094065876545992131370884059645617234469978112000000000000000000000
91	True	135200152767840296255166568759495142147586866476906677791741734597153670771559994765685283954750449427751168336768008192000000000000000000000
92	True	12438414054641307255475324325873553077577991715875414356840239582938137710983519518443046123837041347353107486982656753664000000000000000000000
93	True	1156772507081641574759205162306240436214753229576413535186142281213246807121467315215203289516844845303838996289387078090752000000000000000000000
94	True	108736615665674308027365285256786601004186803580182872307497374434045199869417927630229109214583415458560865651202385340530688000000000000000000000
95	True	10329978488239059262599702099394727095397746340117372869212250571234293987594703124871765375385424468563282236864226607350415360000000000000000000000
96	True	991677934870949689209571401541893801158183648651267795444376054838492222809091499987689476037000748982075094738965754305639874560000000000000000000000
97	True	96192759682482119853328425949563698712343813919172976158104477319333745612481875498805879175589072651261284189679678167647067832320000000000000000000000
98	True	9426890448883247745626185743057242473809693764078951663494238777294707070023223798882976159207729119823605850588608460429412647567360000000000000000000000
99	True	933262154439441526816992388562667004907159682643816214685929638952175999932299156089414639761565182862536979208272237582511852109168640000000000000000000000
100	True	93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
101	True	9425947759838359420851623124482936749562312794702543768327889353416977599316221476503087861591808346911623490003549599583369706302603264000000000000000000000000
102	True	961446671503512660926865558697259548455355905059659464369444714048531715130254590603314961882364451384985595980362059157503710042865532928000000000000000000000000
103	True	99029007164861804075467152545817733490901658221144924830052805546998766658416222832141441073883538492653516385977292093222882134415149891584000000000000000000000000
104	True	10299016745145627623848583864765044283053772454999072182325491776887871732475287174542709871683888003235965704141638377695179741979175588724736000000000000000000000000
105	True	1081396758240290900504101305800329649720646107774902579144176636573226531909905153326984536526808240339776398934872029657993872907813436816097280000000000000000000000000
106	True	114628056373470835453434738414834942870388487424139673389282723476762012382449946252660360871841673476016298287096435143747350528228224302506311680000000000000000000000000
107	True	12265202031961379393517517010387338887131568154382945052653251412013535324922144249034658613287059061933743916719318560380966506520420000368175349760000000000000000000000000
108	True	1324641819451828974499891837121832599810209360673358065686551152497461815091591578895743130235002378688844343005686404521144382704205360039762937774080000000000000000000000000
109	True	144385958320249358220488210246279753379312820313396029159834075622223337844983482099636001195615259277084033387619818092804737714758384244334160217374720000000000000000000000000
110	True	15882455415227429404253703127090772871724410234473563207581748318444567162948183030959960131517678520479243672638179990208521148623422266876757623911219200000000000000000000000000
111	True	1762952551090244663872161047107075788761409536026565516041574063347346955087248316436555574598462315773196047662837978913145847497199871623320096254145331200000000000000000000000000
112	True	197450685722107402353682037275992488341277868034975337796656295094902858969771811440894224355027779366597957338237853638272334919686385621811850780464277094400000000000000000000000000
113	True	22311927486598136465966070212187151182564399087952213171022161345724023063584214692821047352118139068425569179220877461124773845924561575264739138192463311667200000000000000000000000000
114	True	2543559733472187557120132004189335234812341496026552301496526393412538629248600474981599398141467853800514886431180030568224218435400019580180261753940817530060800000000000000000000000000
115	True	292509369349301569068815180481773552003419272043053514672100535242441942363589054622883930786268803187059211939585703515345785120071002251720730101703194015956992000000000000000000000000000
116	True	33931086844518982011982560935885732032396635556994207701963662088123265314176330336254535971207181169698868584991941607780111073928236261199604691797570505851011072000000000000000000000000000
117	True	3969937160808720895401959629498630647790406360168322301129748464310422041758630649341780708631240196854767624444057168110272995649603642560353748940315749184568295424000000000000000000000000000
118	True	468452584975429065657431236280838416439267950499862031533310318788629800927518416622330123618486343228862579684398745837012213486653229822121742374957258403779058860032000000000000000000000000000
119	True	55745857612076058813234317117419771556272886109483581752463927935846946310374691578057284710599874844234646982443450754604453404911734348832487342619913750049708004343808000000000000000000000000000
120	True	6689502913449127057588118054090372586752746333138029810295671352301633557244962989366874165271984981308157637893214090552534408589408121859898481114389650005964960521256960000000000000000000000000000
121	True	09429852527344373968162284544935082997082306309701607045776233628497660426640521713391773997910182738287074185078904956856663439318382745047716214841147650721760223072092160000000000000000000000000000
122	True	50442008336013624115798714482080125644041369783596059584700502676714572050143649033796427745042294071023050579626404736512939596842694895821378210620013388054747214795243520000000000000000000000000000
123	True	04367025329675766243241881295855454217088483382315328918161829235892362167668831156960612640202170735835221294047782591091570411651472186029519906261646730733907419814952960000000000000000000000000000
124	True	41511140879795014161993280686076322918971939407100785852066825250652908790935063463115967385069171243567440461925041295354731044782551067660468376444194611004520057054167040000000000000000000000000000
125	True	88892609974376770249160085759540364871492425887598231508353156331613598866882932889495923133646405445930057740630161919341380597818883457558547055524326375565007131770880000000000000000000000000000000
126	True	00468856771473051394170805702085973808045661837377170052497697783313457227249544076486314839447086187187275319400401837013955325179315652376928996065123321190898603130880000000000000000000000000000000
127	True	59544809977077527059692324164918673621799053346900596667207618480809067860692097713761984609779945772783965563851033300772326297773087851869982500270661791244122597621760000000000000000000000000000000
128	True	21735677065923463640617493109590223590278828403276373402575165543560686168588507361534030051833058916347592172932262498857766114955245039357760034644709279247692495585280000000000000000000000000000000
129	True	03902341504126809639656611137138843145968864022652168932196355119328515747917449637889876686464600208839390308261862352651828829226610077151044469167497022952331930501120000000000000000000000000000000
130	True	07304395536485253155359447828049608975952322944781961185526165512707047229268452925683969240398027149120740074042105844737747799459310029635780991774612983803150965145600000000000000000000000000000000
131	True	56875815279568163352087665474498775849754305766436915303927682164623187034167333264599970492141556534816949699515865660644961729169613882287309922474300878212776434073600000000000000000000000000000000
132	True	07607616902997562475571842633838412167568361169672820118454045730260688510087990927196104962685462595837360336094267205134948250389032461924909766607715924086489297715200000000000000000000000000000000
133	True	11813048098675809251055070300508818286592035566485075754388082124671571841702793317081960037166525246368924700537538282948117301741317436012998958826217903503076596121600000000000000000000000000000000
134	True	82948445222558439641379420268181650403332765909000151088003004705990626788174304488982644980314383013435909872030129915047718433336536425741860482713199069412263880294400000000000000000000000000000000
135	True	98040105045389351586221736204522804449923397715020396880405635308734616403531106012657072342441706813847832724067538531441988500432417475151165166281874370655623839744000000000000000000000000000000000
136	True	33454286172951815726156123815101405189582089242773975735166401987907830880230417721361838572072126683305250473185240276110436058808776620558462614334914409164842205184000000000000000000000000000000000
137	True	83237205694398754483388962668892510972746226260034675717797072343372830591567227826571884373881355612819314826377917827129740056802397016509378163883274055583382110208000000000000000000000000000000000
138	True	86734385827028118707676848307166514238979223884785249055995983385450621636277440066920043595627074569065446040152660143904127838730788278294186615891819670506731208704000000000000000000000000000000000
139	True	56079629956908500367081914696145479218112119985149618783441690577636407442564169301886059792163365100096999581219760002673769583579570682891939608962934200435638009856000000000000000000000000000000000
140	True	51148193967190051391468057460367090535696797920946629681836680869097041958983702264048370902871114013579941370766400374327741701139895604871545254810788060989321379840000000000000000000000000000000000
141	True	11895349373797246196996101911759765533248506853474785138972002542682916216702019230820297304827075914771733278062452780211579860725280286887880928321116599494314557440000000000000000000000000000000000
142	True	89139611079208959973446471469886705721287973193419489734024361060974102771686730776482217285444779897586125484868294790044340222989800738079091821598557128192667156480000000000000000000000000000000000
143	True	46964384326881276202845420193798918144180166658987031965483631719296696351202501036957071818603525354815944336166154976340651887541505545310130488593669331551403376640000000000000000000000000000000000
144	True	62871343070903773209740507907044212761943998894132603029642967578724274573160149321818341878907651093495984407926316593053871805976798524658790357488383743402086236160000000000000000000000000000000000
145	True	16344745281047115412373646521410850481879839649227439298230298915019813108221651663659572441609408556917739149315905992811411866635786075524601835815642793302504243200000000000000000000000000000000000
146	True	86332811032878850206552392125984170354456588787206137541623641592892713800361142894297576474973649309989915800122274950466132528824767026591868029083847822165619507200000000000000000000000000000000000
147	True	90923221833190980363201642519673042105118551719302218618675314155228928653088005461743741821126448568517622617974417718521481737240752909004600275325629858346067558400000000000000000000000000000000000
148	True	56636831312265093753843092911610231557545654456728355563946494973881440657024808338073789526714388140608147460213822341179297111631430532680840748193219035217998643200000000000000000000000000000000000
149	True	38887865527498969322620843829924502074302514052524979028027751108334657896696442372994639480443832950613971571859528835715269633083149369445271480789636247481797836800000000000000000000000000000000000
150	True	33179829124845398393126574488675311145377107878746854204162666250198684504466355949195922066574942592095735778929325357290444962472405416790722118445437122269675520000000000000000000000000000000000000
151	True	10154197851655157362112747789971982951943289690774984828562603780001360174419748328584232052816331406456102618328128950857189333333217935399039885261005462721003520000000000000000000000000000000000000
152	True	43438073451583919041137664075741408695380032997797693941515774560206746511801745944803272028082373781327597985875600530292778666649126180654062559672830333592535040000000000000000000000000000000000000
153	True	46025238092339613294062603588435530393145048663047173051913507711632216305667129554900620296603188543122491838966881134795135997316305640071571629943041039657861120000000000000000000000000000000000000
154	True	87886666220300447285640952619071680544337494109264649994680187591361311072737951454695525676891035640863743200899694758450943586711068571022031011228320107310612480000000000000000000000000000000000000
155	True	22433264146569329274347655956110484372311586936020749175429076661003216274382475477806479918110524333880196139452687559896255940215628508414806740389616633144934400000000000000000000000000000000000000
156	True	99589206864815366798234329153235562080607562019236871366935959116501738803666174537810867225241796085310597754619259343815926673638047312709851500780194770609766400000000000000000000000000000000000000
157	True	35505477776012587322789677057983246655387237020188804608945581290772992175589402436306154362961985393763847475223716979100487761173428095446685622490578985733324800000000000000000000000000000000000000
158	True	09865488609988797000768975161352971551183449189831128213401843942132763743125584936372389347993692214687901085347282697877066265401639080576328353511479745865318400000000000000000000000000000000000000
159	True	68612688988218723122267050655122476638168421183149385930893186799109435156968004883209906330997062135376272570217948962453536198860613811636208208325279592585625600000000000000000000000000000000000000
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992	False	00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
993	False	00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
994	False	00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
995	False	00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
996	False	00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
997	False	00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
998	False	00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
999	False	00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1000	False	00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

If you would like to tinker with the code that generated this result, here it is:

from math import factorial
import sys

import pandas as pd

sys.set_int_max_str_digits(1_000_000)

k = 1000 + 1
m = 200

def is_pandigital(num):
	s = str(num)
	for i in range(10):
		if str(i) not in s:
			return False
	return True

results = []

for n in range(k):
	fact_n = factorial(n)
	fact_str = str(fact_n)[::-1][:m][::-1]

	results.append([n, is_pandigital(fact_str), fact_str])


df = pd.DataFrame(results, columns=["n", f"last_{m}_is_pandigital",  f"last_{m}"])

print(df.to_markdown(index=False))

Happy pattern hunting!