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 Product of 100 factorials (Posted on 2017-09-07)
Show that (1!*2!*3!*4!* ... 100!)/50! is a square number.

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 when it works -- computer exploration | Comment 3 of 8 |
The perfect square condition apparently holds whenever the highest factorial used is the factorial of a multiple of 4 (as is the 100 in the puzzle), and it's divided by the factorial of half of that number (such as the 50 in the puzzle). The proof may be more accessible for smaller numbers like 8 and 4 or 12 and 6, and then generalize.

Each set below shows n; the product of the factorials through that number; that product divided by (n/2)! ; and the square root of that number.  Only the square root has a decimal point, present even if it's .0, indicating the number itself was a perfect square.

4   288   144   12.0

6   24883200   4147200   2036.4675298172568702743

8   5056584744960000   210691031040000   14515200.0

10   6658606584104736522240000000   55488388200872804352000000   7449052839178.468624698672844922

12   127313963299399416749559771247411200000000000   176824949026943634374388571176960000000000   420505587390873600000.0

14   69113789582492712943486800506462734562847413501952000000000000000   13713053488589823996723571529060066381517343948800000000000000   3703114025869285056779734863702.9235841031199276647

16   1890966832292234727042877370627225068196418587883634153182519380410368000000000000000000000   46898978975501853349277712565159351889792127675685370862661690982400000000000000000000   6848282921689337839624757371207680000000000.0

18   430619256499771538211559864037929484578612331960375516802353602787393292715313683117164095078
4000000000000000000000000000
11866712315359665404860005071591972127937950065045621605003130588276931567331175129992396800000000
000000000000000000
3444809474464395600510407473092450940910140021303822989193.704579956292314518

20   127442031238161076418723266959124572809405357648039968146460747862108601584519287331184734157
1189522511934190056991170232320000000000000000000000000000000000
35119607373831866297046755665543588186013381186078033550060832193041391530125464983240943054761616
0304214668776728166400000000000000000000000000000000
592617982969061328644755583860005865281724398591341934673920000000000000000.0

22   731851873447222516859171290615963599944368791704233315972207912248730455032489228114819565481
34913475324429128068839519292427638045690327208977109451014144000000000000000000000000000000000000
000000
18334432455688394782627146730598735368175023842197603915449332417646967067312240162408298397700946
33721250930136404709778650283540907345458778687406080000000000000000000000000000000000000000
42818725408036604450142362971549656456333234665343566420855401315673874704848128599680645038707.02
77397081939008668

24   117387887594348404514585466358206506971534116135679783575233199396355612515468932218157509121
78829716545781568442865877388312912245767262528206470266756517534234506549170924615363740197848088
57600000000000000000000000000000000000000000000000000
24506784026263879810544571533415860609136611680562190935319046824970023589789456281181004222488671
67989789923132379072927587906229492190115483219735958401294324383582261713659278745665536000000000
000000000000000000000000000000000000000
15654642770202033521247987500083042299747510921611551355879583161749951538029431491604711997440000
00000000000000000000.0

26   734324457583597977336142639461562552924008065191130382711801375699858850713343330791066992451
36347067992929024463908068798227802297751609247714136900829761955977081449583014705851127046762891
69327783424988568398136986317713011480985600000000000000000000000000000000000000000000000000000000
000000
11792548654785254247683621668030441666808115771688612035980374044998514389310267452311496895134242
53665459556911451268458878887995648763679217421867304984141950787790038906160484498896274168406395
524393460925712362641589010432000000000000000000000000000000000000000000000000000000000000
34340280509607451731153725404726898656350575085028337804568573220492191077572706894288945712019329
423283454008856657037565573969214825971127832.3595229072026941446

28   243787596925694930018437548842458897097201101239636708838086682677428011500739454622482126715
68308430773839473807128215512114800359197934772179842358860234182649345336155492659539815888875032
45453194984991035988383621511512078568896654636670848281113872246551141153179542298296320000000000
0000000000000000000000000000000000000000000000000000000000000000
27964255042165236890814114608667495549304951418873040360544103286739808396329229107725631437437842
81788473444385100337400524180079271608247831754054713592984792221453424805447384109300925943413597
43810279116366573565221649134431404943004511523194562670235792731522727936000000000000000000000000
000000000000000000000000000000000000000000000000
52881239624431306338784489570165242256069709836430946181037806771527771991538574298712742787059218
4127531112454058865824450897778835456000000000000000000000000000000000000.0

30   571755698232823096732988919122641191801693387811183323217432917650763212540081441313968722920
78960659959192198325482132911362592729947392123616011917207641789593790073387400532018296722819386
98507634634093465979526859113904220996903489507321250439504675693053954628946505956131558297025777
24817000254015466341008537985769810165760000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000
43723094389873603206772415617359656911291036929706289327331444463302595875312090028973384520815935
01109229679290118358648700116811492583459708526757420364286714671620250600532686918638357786254710
49581212978702319173936267306341887668590237386257272320848385242237075530561759206883607416341754
73893972927237265274437632000000000000000000000000000000000000000000000000000000000000000000000000
000000000000
66123440919142738761427621913484709223981773827988104165368437319093927280977907264702955884967724
38108153945445129923165966228455895695184446772711109464652036880681840076833099993245320892412543
344344.3392735976379136939

32   123709775111139644143701308630123507742393581208810389735987207623961953733718650489220817845
00864885892675877720576872849325715592661382917260841250206640434997262948210592804497853876011877
82749436827614617951080711761835420318755984863476083567158644187596578616112029896886114905541029
69541783012283632022364595155234596809986376570773892387693554911390068128193417654398113074380800
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
0
59126806593843305789880954440966141456858461766153157163505712123108786883841525718245944202165783
78083589096987913542655619730129491506805529146673141145695281601896957592199903973855750779418950
86542874820820396660746620321993818028663595058115778677262601098590218768198379049222109661753067
36900632795774106564098396110036961040561386672941430990730881093923927769481216000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
76893957235821402216147893786415565756660953510259442010496946576754456235628883715561946368441290
98373244613680503377163337987301454846163406716885594648249726621314376992573665203716096000000000
0000000000000000000000000000000000000000.0

These being the first 32 lines from

4   kill "pr100fct.txt"
5   open "pr100fct.txt" for output as #2
10   F=1
20   for I=2 to 60
30    F=F*!(I)
40    if I@2=0 and I>2 then
50      :print I,F,F//!(I//2),sqrt(F//!(I//2)):print
51      :print #2,I,F,F//!(I//2),sqrt(F//!(I//2)):print #2,
60   next
70   close #2

Edited on September 8, 2017, 6:59 am
 Posted by Charlie on 2017-09-07 10:53:24

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