Comparing $2013!$ and $1007^{2013}$

I have to compare the following two numbers:

$$2013! \text{ and } 1007^{2013}$$

where $n! = 1 \times 2 \times \cdots \times (n-1) \times n$.

I tried in different ways to group the $1 \times 2 \times \cdots \times 2012 \times 2013$ to obtain some kind of association with the $1007$ from $1007^{2013}$ but no luck.

Is there any standard approach for this kind of problem?

• Do you know the AM/GM intequality - that is, the geometric mean of a set of numbers is always less than or equal to the arithmetic mean? Mar 8, 2013 at 15:43

$1\times 2\times\ldots\times 2012\times 2013$

$=(1007-1006)\times(1007-1005)\times\ldots\times (1007+1005)\times(1007+1006)$

$=(1007^2-1006^2)\times (1007^2-1005^2)\times\ldots\times (1007^2-1^2)\times 1007$

$<(1007^2)^{1006}\times 1007$

$=1007^{2013}$

• Thank you! All answers were helpful but I chose this one because it is really clear and in this case I needed a solution which does not require advanced mathematical knowledge. Thanks again! Mar 8, 2013 at 16:02
• In general, the AM-GM says that $n!\le\left(\frac{n+1}{2}\right)^n$
– robjohn
Mar 8, 2013 at 16:17
• @robjohn that's good to know. Thank you! Mar 8, 2013 at 17:49

You can just take the inequality between the arithmetic and geometric means:

$$\sqrt[n]{a_1a_2\dots a_n}\leq \frac{a_1+a_2\dots+a_n}{n}$$

with $a_i=i$ and $n=2013$. Then $\sqrt[2013]{2013!} \leq 1007$.

• I just noticed that this is the justification for my comment to Jacob Black's answer :-)
– robjohn
Mar 8, 2013 at 20:40
• Yeah, I originally had the same asnwer as most of the rest, but then I saw the AM/GM relationship made it all crystal clear. Most of the "direct" proofs using $(1007-i)(1007+i)<1007^2$ are just using the basic case of AM/GM with $n=2$. :) @robjohn Mar 8, 2013 at 20:54

Use Stirling’s approximation for the factorial:

$$n!\sim\sqrt{2\pi n}\left(\frac{n}e\right)^n\;,$$

so $$2013!\approx 112.46356(740.54132)^{2013}<1007^{2013}\;.$$

For a less computational approach, look at their ratio:

\begin{align*}\frac{2013!}{{1007}^{2013}}&=\frac{2013\cdot2012\cdot\ldots\cdot2\cdot1}{1007\cdot1007\cdot\ldots\cdot1007\cdot1007}\\\\ &=\underbrace{\frac{2013}{1007}\cdot\frac{2012}{1007}\cdot\frac{2011}{1007}\cdot\ldots\cdot\frac{1008}{1007}}_{1006\text{ factors}}\cdot\frac{1007}{1007}\cdot\underbrace{\frac{1006}{1007}\ldots\cdot\frac2{1007}\cdot\frac1{1007}}_{1006\text{ factors}}\;. \end{align*}

The $1006$ factors on the left are between $1$ and $2$; the factors on the right are between $1$ and $\frac1{1007}$. Try matching them up.

The number you used is still small enough to be calculated directly, so I used Python which have unlimited precision integer arithmetic to get the answer:

>>> math.factorial(2013) < 1007**2013
True


$2013!$ is a 5779-digit number:

2842867247059648714320882908184358975298425936765406743563017882090099387783580447012935263746562335479064550804211732063022834149625421992014144514533688428601587813569194324989561917151782797873121680666949119475158973400795970620477769770891025873157955892369624843826017526081184552011934374105097315782048120736085815132954887354715539226256291405434589203833779899000588701510318939043247864678134992934379533230544528487035914118945693626528290368591437206653279522877509212788210737957582469711071411021659870559714136474522599718501409090834918926679268116898270673098203451361899317085587433326969579362334919786008994344217419373006643085721568389423052059764681667173457445382263982941073416605568659124787280831190799962707336132841922714527968274349029365670003854224300000354656476434833237074182399956043794238102817592695696938568409658563414032484092565521956662668178041307970779320324765661689930554915530050651296122418092217569587585513016160316926502774945085654600201428197370411755207529508851026315971786389387156555114178249936514939656226821498735997768211690336174306416146685126970909294083138552874100725271266747942592189688081230332253100251403373027228801998127490992546175714650004281060557576018602691444874370276588961454729805801934914327058639037910641855479898434777131761105855951851430766729127680477612727441963181686830328681323514755070052559807855504808093205862574423306499683388729703624875824538600644082634581722622520635413894047216544019784438604398781448535720783960899315838792367983940191638537067898701245056584637806237613648468018169610742424883398829055732890553015662215573197318332952206241281889197604097179798088111804842744903817885667786672979352296770176797979680935970290662288862203487507979401519096847866678683866374334376409418963307236435361004707445941188594507325969642135133107415655296396624159611926210964204186787681300922165759386829935051758089290240223443538483111578632645575815706664545129018761043826893436452374686888469782790455363052163042518632450978515011042593378534376399702189202730567458796342736238859771708958480074038980187180566104679718225155832885875853654917359185418511438742450543999277408460122783085081722502691860182351628742942469052052715157975688543334095817706215356800313073446341193079209893599903066070195526563109494839115954474386835501264382220444236657418730177384297639296031881749231727029032121673951831434276019390206305794145748383191826612485618071270887347907256258516351143901013690471344368527519645321897619683156486225698317847429747071787435782505800084184225460708682095870996428539068134492961102165074890654386785683264530894831563132549697850639084675795153902191945156201050737533340297121449992322830373504638815285378819787554773575857714346271171627666997528270237625481717740484365952887157175985255192895785552274978335022571528293668119818363728825071916771945692681017826150901126099177461980958354110093788032189674991888570154690670597564487340888355272446828142492550508568505273063085689193317985522452127020856295149656879145397271909956463955318885551026529470343326333517019087867518976826084475341789777460562332134901190219314055148808405315658894204410944546182332408944358895866945504173885656268982293387670502772134839209278543053623115562418454749334700318355178628414168339732521747171855166109477719366226966982601829031348339568014756885616782225353731679589010642963026728648636517643953417223323299231306423925974998506982248088188586105779494525593610906456801237689150909962837142846451849955648319845974068370644068187479029156569762901273685536943757280003362988100863219269554589016589760560713983742575493275626526617700401513433557529843197510279124020494964257680880911644344603495723958279131978382862453869345658201667502410202572628757335499674300863432592256152060213732587660770868535439645357581075380700393727832817791761732689216175521623731997708698450168187892421709215550480506135272662476910084841930084011031354220700664181552094272535475251690999640528044608914938782665232028970687185403776703990325979508532509493002666906494481541013473948623057439067716354970459715223485500393463339996144504094685302690606100003782316415853025878871684344019606419775272528889669587663352056426642112160119700550423526808274366819908984680988815165676455989729650205942158024441753903462713582925148615281566818726873632617486647784105131818236742851031943060670450995458031472353966387062287071892889526379622030471150370864215268673980706329585427011992451333823961336244992035822313120108408842120520563544476799611543060018205099439432597579853313019154300707372638257691230423084283981806019389567318013796940093147068650783769412807683958066857778735252857988431592811150528499154617178393545996237667501439149410415620227944238095379762968189409689341536890547638203388318768592606767408394815503891306346323652116455703172953351201933588474955527563220129086963377342371823922961841428147215856281545446983868005765000965710748347784855863778139511179159754686588299301850513114088473086011993200873704826365083841064351095378526835498963203426650033008917628965391592199416177660939562162051582018360903009409052540002566924884243359804526539615984308373031809595967024782906756331879790638545240064000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

and $1007^{2013}$ is a 6046-digit number:

12540770622340186642947856510793531886611912530777993058137911797578274638051861071232734697741620539036641265679212148809320334050510282436155553971915019593642601063230039285993210027316833229494019996552105028093917449384199390180627383327163664028628804352157574535269851978605241188192948561279832554499491936504664692559203076376719893471929908965648318021933036626374716898582548817753082949310563145935588658594122778469592203160059819913909820436184842607015273309981919874676707341507357662169232949073263880712042750658289370026048283334154877951521224477815038328455943600266175073512832249210166111988314930309304821959817581310558181836897479958903656656514762791074248152878686368693611099285066813477437771199117201326605447860462184325655714417886214306040804849382962033945680601679647853297347666822824453487097963119304626348810136207993650535865089187425317239955805145063258550512978367416014840538660072703157026590268795007278729829253999203060048785691953718047978785881698065677434716076521849039356546823668783369224020958737250660234554956739716809177968709319891768695573394838084371253037444242858895372290590446426196483469142832255798411673483749923961398830939535978830220266139409099479632602009340554505181898366032556799616519322340437892078308105009833157291531950868737961019457462715711059697605605584746734624571928423070253822044089781399853169426349766426960362396941448024944373806333452700555181556618725995616027828315510158543603768523621988960880542941087458463723237448681117793023018648287870342945384311867408054077292212210925380266103074112848724635330747161830392387189344531055031157090747598915495726074757461242709041546172692828724310644187606574049434803767037723948301935425349239802947032004919893998884555192641407250157544387478084907547282281953505422841458851462675189734371421060529653764746973347691680782684395887200638997846783924632124201532716043449511957770998671036230291812561515658633422643580811032376778997425322543070113242978519326057372277576195823313373629372025320377030489943394861333510103957725890977383320780109377420012231386881031913383420167855469002868871364135460841020314781548021485937450118030491172486762685200245901017203022100220814753330497186240660896740623396963257670973204205749212128070531423976841077346923345890195088924368145133802094133334537864233019513171379318911448439419992240669249787013765391668638784366386017011496054610213094995317870078469919339870711396773279210443985308668473738304238872501322875812630566563309665540439804488941791038761806235428421067411763142818094953690022795092210374864965969551738705254605247720560625049968655999533810963121595103986686326363266359249382233899303950092795146084513537407256446693882647520615279445154589341347556262892551088361851211811847715767738467590185362877136208892231689386725146948303729235800404475275414976627466325267089203567783645709897553753482273240202908417062634027964532930811188667938171439657369551726221747404444270311003296926405727557268176450104765828612273085156678023096572131405296866198347587390659767201963996330295350942534449264814668107804442445066512815976933049076518554357950313681286365958396312051603092688036238255084145979160688160078304417245389443869872897581719966860367139772722686460977719333699506892311676532183689726622250339552872134447245871511297460638151519876038608789297433386999083082138467728810780760166542249927744748937804148568614184557585685559281208539965195706721711352321923197643777718659709427260575996667182589184066512230516366684129134747081386608865809852630855464589822967021371423512066965790317995563448246932406250776727096599982867252055090048302831976569032105828395630242322603265927558194733033838615276769988605456876450633187485302834057397938810251134442357170056661712055027132368946638244916956094177953804741644846467185906553024396055755753715751904057531487694836417958996603739674280600406577887373711166328664912565111584138660091947985972717061773223068694903285516075689032920983166665990231408646216170879294842840440791918056727330600815113486979472534110271240954004872458447308011752579836210243314953217865316169220391629135927680054370364810412755363976249077729940161563021909329861737311243274443040402778189987110836924256648020285664514891319221306366363275630584541007747907990302696118647274087073517415065108868012582248621300278192836296644308531850281964753239278256454593247149123692878999651297025872315344133460680780715189588428373139640175330522626040622166563955858432908789411512263277119120066542480002599083434918179137387176236293954422279424266343400365508282860357065716664640191973915025253310996405274935381525732094178654969594130352052515861416045100888347159102227996288036710890962039178922782186371915310643849047358951146681533619733526718663319338810487251868787433028512549288496605489501829806509886788815137244575585003063804517480797026908414269969177462501408209498642124594908511942953262984079591669946781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(though I guess this is not really what you're looking for)

• Better tag this one intuition. Mar 8, 2013 at 21:04

Let $j \in \{1,2,\dots,2013\}$.

When $j$ is small, $1007$ is much larger in size than $j$. [For e.g. $1007 = 1007 \cdot 1$] So for small $j$, the $j$-th factor in $1007^{2013}$ is much larger in size than the $j$-th factor in $2013!$.

But when $j$ is large, $j$ is not that much larger than $1007$. [For e.g. $2013 < 1007 \cdot 2$] So for large $j$, the $j$-th factor in $2013!$ is not that much larger in size than the $j$-th factor in $1007^{2013}$.

[Notice also that $1007$ is the median (midpoint) of $\{1,2,\dots,2013\}$.]

• I upvoted, but you should change term to factor. Mar 8, 2013 at 15:40
• Also, "median" isn't really the key. If $a_1,\dots,a_{2013}$ is a set of positive numbers with mean $1007$ then it is still true that $a_1\dots a_n \leq 1007^{2013}$. Mar 8, 2013 at 15:46
• Thanks Brian & Thomas - have edited my answer accordingly to take in your comments. Mar 9, 2013 at 15:23

Hint: Use the inequality $k(n-k)\le (n/2)^2$. Then you could prove the more general inequality (without Stirling).

$1007^{2013}=(2013-1006)(2012-1005)(2011-1004).....(1008-1)$

$2013!=(1007+1006).(1007+1005).(1007+1004)....(1007-1006).(1007)$

$(1007^2-1006^2).(1007^2-1005^2).(1007^2-1004^2)...(1007^2-1^2).1007<(1007^2)^{1006} .1007$

$<1007^{2013}$

A mere generalization, using Stirling's approximation again:

$$n! \ \text{vs} \bigg(\frac{n}{2} \bigg)^n\\ \bigg(\frac{n}{e}\bigg)^n \sqrt{2 \pi n} \ \text{vs} \ \bigg(\frac{n}{2} \bigg)^n\\ 2 \pi n \ \text{vs} \bigg(\frac{e}{2} \bigg)^{2n}$$ LHS is $f(n)=2 \pi n=O(n)$, RHS is $g(n)=(\frac{e}{2} )^{2n}=O(a^{2n}), \ a>1$, so $f(n)=o(g(n))$. In fact this holds already for $n=6$.

• It's actually $\left(\frac{n+1}2\right)^n$. Mar 8, 2013 at 15:53