快速精确 bigint 阶乘
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【中文标题】快速精确 bigint 阶乘【英文标题】：Fast exact bigint factorial 【发布时间】：2013-08-21 11:26:53 【问题描述】：我有一个定点大数库，想实现没有精度损失的快速阶乘。
在纸上做了一些数学技巧后，我得到了这个公式：
(4N)!=((2N)!).((2N)!). (2N+1).(2N+3).(2N+5)...(4N-1) .(2^N)/(N!)
这已经相当快了，加上一些编程技巧，复杂度接近~ O(log(n))。
说清楚，我目前的实现是这样的：
//---------------------------------------------------------------------------
longnum fact(const DWORD &x,longnum &h) // h return (x>>1)! to speed up computation
if (x==0) h=1; return 1;
if (x==1) h=1; return 1;
if (x==2) h=1; return 2;
if (x==3) h=1; return 6;
if (x==4) h=2; return 24;
int N4,N2,N,i; longnum c,q;
N=(x>>2);
N2=N<<1;
N4=N<<2;
h=fact(N2,q); // get 2N! and N!
c=h*h; for (i=(N2+1)|1;i<=N4;i+=2) c*=i; c/=q; // c= ((2N!)^2)*T1 / N!
for (i=N4+1;i<=x;i++) c*=i; c.round(); c<<=N ; // convert 4N! -> x!, cut off precision losses
for (i=(N2+1)|1,N2=x>>1;i<=N2;i++) h*=i; h.round(); // convert 2N! -> (x/2)!, cut off precision losses
return c;
//---------------------------------------------------------------------------
longnum fact(const DWORD &x)
longnum tmp;
return fact(x,tmp);
//---------------------------------------------------------------------------
现在我的问题：
有没有一种快速获取 N! 的术语： T1 = (2N+1).(2N+3).(2N+5)...(4N-1) ？
已经回答了。
为了清楚起见，我需要提取这个未知术语：
T2 = (4N)! / (((2N)!).((2N)!))
所以：
(4N)! = (((2N)!).((2N)!)).T2
这会很有帮助，因为这样就不需要为阶乘计算 .../(N!)。
T1 术语始终可以整数分解为：
T1 = T2 * N!
最后，它击中了我 :) 我做了一个小程序来分解阶乘的素数，然后突然一切都变得清晰了：
4! = 2!.2!.(2^1).(3^1) = 24
8! = 4!.4!.(2^1).(5^1).(7^1) = 40320
12! = 6!.6!.(2^2).(3^1).(7^1).(11^1) = 479001600
16! = 8!.8!.(2^1).(3^2).(5^1).(11^1).(13^1) = 20922789888000
20! = 10!.10!.(2^2).(11^1).(13^1).(17^1).(19^1) = 2432902008176640000
24! = 12!.12!.(2^2).(7^1).(13^1).(17^1).(19^1).(23^1) = 620448401733239439360000
28! = 14!.14!.(2^3).(3^3).(5^2).(17^1).(19^1).(23^1) = 304888344611713860501504000000
32! = 16!.16!.(2^1).(3^2).(5^1).(17^1).(19^1).(23^1).(29^1).(31^1) = 263130836933693530167218012160000000
36! = 18!.18!.(2^2).(3^1).(5^2).(7^1).(11^1).(19^1).(23^1).(29^1).(31^1) = 371993326789901217467999448150835200000000
40! = 20!.20!.(2^2).(3^2).(5^1).(7^1).(11^1).(13^1).(23^1).(29^1).(31^1).(37^1) = 815915283247897734345611269596115894272000000000
在分析了T2 项的质指数（半阶乘 ^ 2 之后的其余部分）后，我推导出它们的公式：
T2(4N) = multiplication(i=2,3,5,7,11,13,17,...) of ( i ^ sum(j=1,2,3,4,5,...) of (4N/(i^j))-(2N/(i^j)) )
primes <= 4N
直到i^j <= 4N
问题在于 4N/(i^j) 和 2N/(i^j) 的除法必须在整数数学中完成，因此它们不能轻易简化。
所以我还有一个问题：
我怎样才能有效地计算这个：exponent(i) = sum(j=1,2,3,4,5,...) of (N/(i^j))？
i 是任何素数，其中i<=N。这应该很容易。
现在我计算T2(N) 中的素数i 的指数e，就像这样（但这对我来说太复杂了）：
for (e=0,a=N/i,b=(N>>1)/i;(a)||(b);e+=a-b-b,a/=i,b/=i);
...我将尝试将T2 实现到fact(x) 并比较速度...
【问题讨论】：
这段代码看起来很复杂。 O(n) 循环有什么问题？ @CarlNorum 这正是我的想法。 “经过一些方法，我得到了公式 [...]，并且通过一些编程技巧，复杂性接近 O(nlogn)”for(long long int i = 1 ; i <= n ; ++i) n *= i; 典型的 for 循环 O(n) 实现有什么问题？
对不起我的错误应该是 O(log(n)) 这使用 N 的细分来计算 40！它使用20！和10！ , 计算 20!它使用 10！和5！ ... 等等。计算100！在 O(n) 情况下，您只需要 5 次递归而不是 99 次
您的问题可能有一些错字 - 不可能获得 N！从 (2N+1).(2N+3).(2N+5)...(4N-1)，因为所有数字 (2N+x) 都是奇数，而 N!甚至从 N=2 开始；
公式必须正确，因为它们产生正确的结果，(2N+x) 不仅是奇数... (2N+1+2x) 是奇数，...不要忽略 /N！在 (4N)!=((2N)!).((2N)!). (2N+1).(2N+3).(2N+5)...(4N-1) 的末尾.(2^N)/(N!) 这就是你所缺少的，... PS 所有不可被 4 整除的阶乘都是通过将最后 1-3 个数字乘以 ((N>>2)
【参考方案1】：
我认为你想多了，计算阶乘的好处是你可以使用最后的计算来计算新的，所以显然最好的方法是缓存结果，这也将更容易实现比你的解决方案。
我还在另一个问题上看到，您可以通过最少的次数使用大 num 乘法来加快每次运行，这样做的方法是继续乘法，直到达到大 num 的大小，然后开始乘以下一个数字，直到你得到一个大数字。重复此操作，仅在最后将您剩下的所有大数字相乘。
我的解决方案很简单，但与大多数编程问题一样，它已经有了公认的最快解决方案。您可以使用一种称为prime swing 的技术，我还没有尝试了解它，但它遍布整个互联网，因此您应该可以轻松找到它
【讨论】：
这段代码已经兑现了最后两个半阶乘，这大大加快了速度，缓存更多值不是我的实现的选择，因为它需要大量内存，并且查找表的代码也比代码的当前状态。此代码执行 100！通过 5 层递归，与查找表相比并没有那么多......在定点上大 int 乘法也不是问题（只需为较小的数字添加少量 if()），因此对乘数进行排序的代码实际上会减慢速度向下 需要T2的真正原因是减少递归数据（只需要后半个阶乘而不是两个），也不需要除法 @Spektre 我不明白为什么考虑到阶乘的大小增长速度有多快，您的查找表会变慢 1.第一个 1-N 阶乘（这是唯一一个没有减慢但它只切断了几层递归），2. 最后 N 个使用的阶乘（需要比较和搜索比切割递归层更复杂的大数字表以获得更大表和小表是没用的），首先是 c*N 事实（这可能很好，但如果 bignum 的大小不受限制，那么它对于更大的阶乘也无用）...... @Spektre 我真的不明白你想说什么，查找表对于阶乘应该是微不足道的：1.因为你可以有恒定的时间访问（N！在第 N 点）2。由于阶乘增加 非常 快，查找表甚至不会那么大 3. 我没有检查过，但 cmets 中的一些人似乎认为你的公式甚至没有工作【参考方案2】：我有一个解决方案：
(4N!)=((2N!)^2) . mul(i=all primes<=4N) of [i^sum(j=1,2,3,4,5,...4N>=i^j) of [(4N/(i^j))%2]]
T2 的子项总是prime^exponent，其中指数可以用这样的小整数计算：
for (e=0,j=N4;j;e+=j&1,j/=p);
其中e 是指数，p 是质数，N4 是4*N
新方程的代码：
// edit beg:
// Sorry, forget to copy sorted list of all primes up to max n here it is
// end of table is marked with 0
// Primes are in DWORDs so they only 4Byte per number
// so the table is very small compared with lookup table for the same max n!
// and also primes are needed for many other routines in bignum
// can compute n! for n <= max prime in table
DWORD _arithmetics_primes[]=2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,0;
// edit end.
longnum fact(const DWORD &x)
if (x<=4)
if (x==4) return 24;
if (x==3) return 6;
if (x==2) return 2;
if (x==1) return 1;
if (x==0) return 1;
int N4,N2,p,i,j,e; longnum c,pp;
N4=(x>>2)<<2;
N2=N4>>1;
c=fact(N2); c*=c; // c=((2N)!)^2;
for (i=0;;i++) // c*= T2
p=_arithmetics_primes[i];
if (!p) break;
if (p>N4) break;
for (e=0,j=N4;j;e+=j&1,j/=p);
if (e) // c*=p^e
if (p==2) c<<=e;
else for (pp=p;;)
if (int(e&1)) c*=pp;
e>>=1; if (!e) break;
pp*=pp;
for (i=N4+1;i<=x;i++) c*=i; c.round();
return c;
这里是前 128 个阶乘的粗略时间测量，以便您估计实际复杂性。
Fixed point 768.128 bits arithmetics ... 231.36 decimals.
[ 0.001 ms ] 1! = 1
[ 0.000 ms ] 2! = 2
[ 0.000 ms ] 3! = 6
[ 0.000 ms ] 4! = 24
[ 0.006 ms ] 5! = 120
[ 0.006 ms ] 6! = 720
[ 0.007 ms ] 7! = 5040
[ 0.005 ms ] 8! = 40320
[ 0.006 ms ] 9! = 362880
[ 0.007 ms ] 10! = 3628800
[ 0.008 ms ] 11! = 39916800
[ 0.012 ms ] 12! = 479001600
[ 0.013 ms ] 13! = 6227020800
[ 0.014 ms ] 14! = 87178291200
[ 0.016 ms ] 15! = 1307674368000
[ 0.014 ms ] 16! = 20922789888000
[ 0.015 ms ] 17! = 355687428096000
[ 0.017 ms ] 18! = 6402373705728000
[ 0.019 ms ] 19! = 121645100408832000
[ 0.016 ms ] 20! = 2432902008176640000
[ 0.017 ms ] 21! = 51090942171709440000
[ 0.019 ms ] 22! = 1124000727777607680000
[ 0.021 ms ] 23! = 25852016738884976640000
[ 0.023 ms ] 24! = 620448401733239439360000
[ 0.025 ms ] 25! = 15511210043330985984000000
[ 0.027 ms ] 26! = 403291461126605635584000000
[ 0.029 ms ] 27! = 10888869450418352160768000000
[ 0.032 ms ] 28! = 304888344611713860501504000000
[ 0.034 ms ] 29! = 8841761993739701954543616000000
[ 0.037 ms ] 30! = 265252859812191058636308480000000
[ 0.039 ms ] 31! = 8222838654177922817725562880000000
[ 0.034 ms ] 32! = 263130836933693530167218012160000000
[ 0.037 ms ] 33! = 8683317618811886495518194401280000000
[ 0.039 ms ] 34! = 295232799039604140847618609643520000000
[ 0.041 ms ] 35! = 10333147966386144929666651337523200000000
[ 0.039 ms ] 36! = 371993326789901217467999448150835200000000
[ 0.041 ms ] 37! = 13763753091226345046315979581580902400000000
[ 0.044 ms ] 38! = 523022617466601111760007224100074291200000000
[ 0.046 ms ] 39! = 20397882081197443358640281739902897356800000000
[ 0.041 ms ] 40! = 815915283247897734345611269596115894272000000000
[ 0.044 ms ] 41! = 33452526613163807108170062053440751665152000000000
[ 0.046 ms ] 42! = 1405006117752879898543142606244511569936384000000000
[ 0.049 ms ] 43! = 60415263063373835637355132068513997507264512000000000
[ 0.048 ms ] 44! = 2658271574788448768043625811014615890319638528000000000
[ 0.050 ms ] 45! = 119622220865480194561963161495657715064383733760000000000
[ 0.054 ms ] 46! = 5502622159812088949850305428800254892961651752960000000000
[ 0.056 ms ] 47! = 258623241511168180642964355153611979969197632389120000000000
[ 0.056 ms ] 48! = 12413915592536072670862289047373375038521486354677760000000000
[ 0.060 ms ] 49! = 608281864034267560872252163321295376887552831379210240000000000
[ 0.063 ms ] 50! = 30414093201713378043612608166064768844377641568960512000000000000
[ 0.066 ms ] 51! = 1551118753287382280224243016469303211063259720016986112000000000000
[ 0.065 ms ] 52! = 80658175170943878571660636856403766975289505440883277824000000000000
[ 0.069 ms ] 53! = 4274883284060025564298013753389399649690343788366813724672000000000000
[ 0.072 ms ] 54! = 230843697339241380472092742683027581083278564571807941132288000000000000
[ 0.076 ms ] 55! = 12696403353658275925965100847566516959580321051449436762275840000000000000
[ 0.077 ms ] 56! = 710998587804863451854045647463724949736497978881168458687447040000000000000
[ 0.162 ms ] 57! = 40526919504877216755680601905432322134980384796226602145184481280000000000000
[ 0.095 ms ] 58! = 2350561331282878571829474910515074683828862318181142924420699914240000000000000
[ 0.093 ms ] 59! = 138683118545689835737939019720389406345902876772687432540821294940160000000000000
[ 0.089 ms ] 60! = 8320987112741390144276341183223364380754172606361245952449277696409600000000000000
[ 0.093 ms ] 61! = 507580213877224798800856812176625227226004528988036003099405939480985600000000000000
[ 0.098 ms ] 62! = 31469973260387937525653122354950764088012280797258232192163168247821107200000000000000
[ 0.096 ms ] 63! = 1982608315404440064116146708361898137544773690227268628106279599612729753600000000000000
[ 0.090 ms ] 64! = 126886932185884164103433389335161480802865516174545192198801894375214704230400000000000000
[ 0.100 ms ] 65! = 8247650592082470666723170306785496252186258551345437492922123134388955774976000000000000000
[ 0.104 ms ] 66! = 544344939077443064003729240247842752644293064388798874532860126869671081148416000000000000000
[ 0.111 ms ] 67! = 36471110918188685288249859096605464427167635314049524593701628500267962436943872000000000000000
[ 0.100 ms ] 68! = 2480035542436830599600990418569171581047399201355367672371710738018221445712183296000000000000000
[ 0.121 ms ] 69! = 171122452428141311372468338881272839092270544893520369393648040923257279754140647424000000000000000
[ 0.109 ms ] 70! = 11978571669969891796072783721689098736458938142546425857555362864628009582789845319680000000000000000
[ 0.119 ms ] 71! = 850478588567862317521167644239926010288584608120796235886430763388588680378079017697280000000000000000
[ 0.104 ms ] 72! = 61234458376886086861524070385274672740778091784697328983823014963978384987221689274204160000000000000000
[ 0.124 ms ] 73! = 4470115461512684340891257138125051110076800700282905015819080092370422104067183317016903680000000000000000
[ 0.113 ms ] 74! = 330788544151938641225953028221253782145683251820934971170611926835411235700971565459250872320000000000000000
[ 0.118 ms ] 75! = 24809140811395398091946477116594033660926243886570122837795894512655842677572867409443815424000000000000000000
[ 0.118 ms ] 76! = 1885494701666050254987932260861146558230394535379329335672487982961844043495537923117729972224000000000000000000
[ 0.123 ms ] 77! = 145183092028285869634070784086308284983740379224208358846781574688061991349156420080065207861248000000000000000000
[ 0.129 ms ] 78! = 11324281178206297831457521158732046228731749579488251990048962825668835325234200766245086213177344000000000000000000
[ 0.133 ms ] 79! = 894618213078297528685144171539831652069808216779571907213868063227837990693501860533361810841010176000000000000000000
[ 0.121 ms ] 80! = 71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000
[ 0.119 ms ] 81! = 5797126020747367985879734231578109105412357244731625958745865049716390179693892056256184534249745940480000000000000000000
[ 0.131 ms ] 82! = 475364333701284174842138206989404946643813294067993328617160934076743994734899148613007131808479167119360000000000000000000
[ 0.150 ms ] 83! = 39455239697206586511897471180120610571436503407643446275224357528369751562996629334879591940103770870906880000000000000000000
[ 0.141 ms ] 84! = 3314240134565353266999387579130131288000666286242049487118846032383059131291716864129885722968716753156177920000000000000000000
[ 0.148 ms ] 85! = 281710411438055027694947944226061159480056634330574206405101912752560026159795933451040286452340924018275123200000000000000000000
[ 0.154 ms ] 86! = 24227095383672732381765523203441259715284870552429381750838764496720162249742450276789464634901319465571660595200000000000000000000
[ 0.163 ms ] 87! = 2107757298379527717213600518699389595229783738061356212322972511214654115727593174080683423236414793504734471782400000000000000000000
[ 0.211 ms ] 88! = 185482642257398439114796845645546284380220968949399346684421580986889562184028199319100141244804501828416633516851200000000000000000000
[ 0.151 ms ] 89! = 16507955160908461081216919262453619309839666236496541854913520707833171034378509739399912570787600662729080382999756800000000000000000000
[ 0.157 ms ] 90! = 1485715964481761497309522733620825737885569961284688766942216863704985393094065876545992131370884059645617234469978112000000000000000000000
[ 0.166 ms ] 91! = 135200152767840296255166568759495142147586866476906677791741734597153670771559994765685283954750449427751168336768008192000000000000000000000
[ 0.161 ms ] 92! = 12438414054641307255475324325873553077577991715875414356840239582938137710983519518443046123837041347353107486982656753664000000000000000000000
[ 0.169 ms ] 93! = 1156772507081641574759205162306240436214753229576413535186142281213246807121467315215203289516844845303838996289387078090752000000000000000000000
[ 0.173 ms ] 94! = 108736615665674308027365285256786601004186803580182872307497374434045199869417927630229109214583415458560865651202385340530688000000000000000000000
[ 0.188 ms ] 95! = 10329978488239059262599702099394727095397746340117372869212250571234293987594703124871765375385424468563282236864226607350415360000000000000000000000
[ 0.181 ms ] 96! = 991677934870949689209571401541893801158183648651267795444376054838492222809091499987689476037000748982075094738965754305639874560000000000000000000000
[ 0.187 ms ] 97! = 96192759682482119853328425949563698712343813919172976158104477319333745612481875498805879175589072651261284189679678167647067832320000000000000000000000
[ 0.194 ms ] 98! = 9426890448883247745626185743057242473809693764078951663494238777294707070023223798882976159207729119823605850588608460429412647567360000000000000000000000
[ 0.201 ms ] 99! = 933262154439441526816992388562667004907159682643816214685929638952175999932299156089414639761565182862536979208272237582511852109168640000000000000000000000
[ 0.185 ms ] 100! = 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
[ 0.191 ms ] 101! = 9425947759838359420851623124482936749562312794702543768327889353416977599316221476503087861591808346911623490003549599583369706302603264000000000000000000000000
[ 0.202 ms ] 102! = 961446671503512660926865558697259548455355905059659464369444714048531715130254590603314961882364451384985595980362059157503710042865532928000000000000000000000000
[ 0.207 ms ] 103! = 99029007164861804075467152545817733490901658221144924830052805546998766658416222832141441073883538492653516385977292093222882134415149891584000000000000000000000000
[ 0.242 ms ] 104! = 10299016745145627623848583864765044283053772454999072182325491776887871732475287174542709871683888003235965704141638377695179741979175588724736000000000000000000000000
[ 0.210 ms ] 105! = 1081396758240290900504101305800329649720646107774902579144176636573226531909905153326984536526808240339776398934872029657993872907813436816097280000000000000000000000000
[ 0.215 ms ] 106! = 114628056373470835453434738414834942870388487424139673389282723476762012382449946252660360871841673476016298287096435143747350528228224302506311680000000000000000000000000
[ 0.221 ms ] 107! = 12265202031961379393517517010387338887131568154382945052653251412013535324922144249034658613287059061933743916719318560380966506520420000368175349760000000000000000000000000
[ 0.217 ms ] 108! = 1324641819451828974499891837121832599810209360673358065686551152497461815091591578895743130235002378688844343005686404521144382704205360039762937774080000000000000000000000000
[ 0.226 ms ] 109! = 144385958320249358220488210246279753379312820313396029159834075622223337844983482099636001195615259277084033387619818092804737714758384244334160217374720000000000000000000000000
[ 0.232 ms ] 110! = 15882455415227429404253703127090772871724410234473563207581748318444567162948183030959960131517678520479243672638179990208521148623422266876757623911219200000000000000000000000000
[ 0.240 ms ] 111! = 1762952551090244663872161047107075788761409536026565516041574063347346955087248316436555574598462315773196047662837978913145847497199871623320096254145331200000000000000000000000000
[ 0.213 ms ] 112! = 197450685722107402353682037275992488341277868034975337796656295094902858969771811440894224355027779366597957338237853638272334919686385621811850780464277094400000000000000000000000000
[ 0.231 ms ] 113! = 22311927486598136465966070212187151182564399087952213171022161345724023063584214692821047352118139068425569179220877461124773845924561575264739138192463311667200000000000000000000000000
[ 0.240 ms ] 114! = 2543559733472187557120132004189335234812341496026552301496526393412538629248600474981599398141467853800514886431180030568224218435400019580180261753940817530060800000000000000000000000000
[ 0.252 ms ] 115! = 292509369349301569068815180481773552003419272043053514672100535242441942363589054622883930786268803187059211939585703515345785120071002251720730101703194015956992000000000000000000000000000
[ 0.248 ms ] 116! = 33931086844518982011982560935885732032396635556994207701963662088123265314176330336254535971207181169698868584991941607780111073928236261199604691797570505851011072000000000000000000000000000
[ 0.598 ms ] 117! = 3969937160808720895401959629498630647790406360168322301129748464310422041758630649341780708631240196854767624444057168110272995649603642560353748940315749184568295424000000000000000000000000000
[ 0.259 ms ] 118! = 468452584975429065657431236280838416439267950499862031533310318788629800927518416622330123618486343228862579684398745837012213486653229822121742374957258403779058860032000000000000000000000000000
[ 0.261 ms ] 119! = 55745857612076058813234317117419771556272886109483581752463927935846946310374691578057284710599874844234646982443450754604453404911734348832487342619913750049708004343808000000000000000000000000000
[ 0.254 ms ] 120! = 6689502913449127057588118054090372586752746333138029810295671352301633557244962989366874165271984981308157637893214090552534408589408121859898481114389650005964960521256960000000000000000000000000000
[ 0.263 ms ] 121! = 809429852527344373968162284544935082997082306309701607045776233628497660426640521713391773997910182738287074185078904956856663439318382745047716214841147650721760223072092160000000000000000000000000000
[ 0.270 ms ] 122! = 98750442008336013624115798714482080125644041369783596059584700502676714572050143649033796427745042294071023050579626404736512939596842694895821378210620013388054747214795243520000000000000000000000000000
[ 0.281 ms ] 123! = 12146304367025329675766243241881295855454217088483382315328918161829235892362167668831156960612640202170735835221294047782591091570411651472186029519906261646730733907419814952960000000000000000000000000000
[ 0.290 ms ] 124! = 1506141741511140879795014161993280686076322918971939407100785852066825250652908790935063463115967385069171243567440461925041295354731044782551067660468376444194611004520057054167040000000000000000000000000000
[ 0.322 ms ] 125! = 188267717688892609974376770249160085759540364871492425887598231508353156331613598866882932889495923133646405445930057740630161919341380597818883457558547055524326375565007131770880000000000000000000000000000000
[ 0.303 ms ] 126! = 23721732428800468856771473051394170805702085973808045661837377170052497697783313457227249544076486314839447086187187275319400401837013955325179315652376928996065123321190898603130880000000000000000000000000000000
[ 0.313 ms ] 127! = 3012660018457659544809977077527059692324164918673621799053346900596667207618480809067860692097713761984609779945772783965563851033300772326297773087851869982500270661791244122597621760000000000000000000000000000000
[ 0.307 ms ] 128! = 385620482362580421735677065923463640617493109590223590278828403276373402575165543560686168588507361534030051833058916347592172932262498857766114955245039357760034644709279247692495585280000000000000000000000000000000
refernce 128! = 385620482362580421735677065923463640617493109590223590278828403276373402575165543560686168588507361534030051833058916347592172932262498857766114955245039357760034644709279247692495585280000000000000000000000000000000
我的测量显示N! 使用
2.2N 的最大值为快速低级长操作 (+,-,<<,>>)
略小于N/2的长乘法，但大部分在大小上都很方便，加速了乘法，所以测量的时间与明显的O(N/2*O(multiplication))不匹配。使用analysis on the times 后，观察到的复杂性与O(N.log^2(N)) 最匹配，但错误仍然太大，因此很可能是一些稍微奇怪的函数...
我也尝试过仅将阶乘作为素数的非递归乘法（类似于T2 术语），但结果要慢得多。
PS： 问题中发布的代码也可以100% 工作，但比新代码慢（即使它使用较少的乘法 - 因为递归需要更多内存且未优化乘法器顺序） .
【讨论】：
这是摆动素数吗？ 不，但素数分解的指数看起来与该算法非常相似（据我查看您的链接）。以上是关于快速精确 bigint 阶乘的主要内容，如果未能解决你的问题，请参考以下文章
MySQL INT,TINYINT,SMALLINT,BIGINT用法和应用
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