# Largest number using 1, 2, 3, 4 [closed]

Find the largest number using digits 1,2,3,4. Allowed operations are addition, subtraction, multiplication, division, exponents, brackets and decimal point. No other digits besides one of each of 1, 2, 3, and 4 are allowed.

• Concatenation allowed (e.g. $43^{21}$)?
– z100
Commented Dec 25, 2022 at 22:45
• What kind of operation is "decimal point"? Division by 10? Concatenation with a decimal point in between?... Commented Dec 26, 2022 at 18:44

$$\large{.3^{-.2^{-.1^{-4}}}}$$

which is approximately equal to

8.2868580201393671 × 10262086317397329836950303705682257163703149058949801218729493738441081598107528212748648366556650639675637494834651527009111141529869553576563913474335144955245264754635782473137984138347137952222638587964313205619452868837383718224634189273547166482909758564678494081744342572499686332340283615712581599021968814570900261444208208929835178424692667808106080151784111892423283390244145242205430052948581301571055543827374667243547881546403168164256188568714616762528088966293537298281238136576979323677929910645315459531637368751080803173205102267159826018600957959640410062625909165800472087302204963540888240564840302750650240671426732699878134259276410912317928891377617777223758778463382382989565963629022860722966911068954608859273028192166051552188580748979849856353251278350155772482582741438334143652808512554106991133426578977315310016893382680601367236364084069749470853443372254750811042507482354057432410918072658245044727765974298088591966668845706028688485051186137975830957602719621770168245055731886715877988832778550404172876713201400626562113916859758503503572420850070382232653128175849958311906700033232287139899003408154625886599216875941009235625509735126594114998258345932048819865655098527930205860927170520898038642117761923498823378971868312904039581254514465902510080892393070829681344739389363741514397040092807009060910726875881384453141825033744104371971395737958626485122501355695379269195842561891300017940444007250843146820361922480751344116392255182941972803187740676380024972345500168720227172189949951860691398299398246182652471366029534374277502048202948135938069081003223661885217857165791753402969255926344169840499123321398708367849182966199587193351349701644514591422555770850315692644641571136329357676395384520945095819221313693113684815506633593297676946168808642863322194517958045547364131780483725049561364697828657935781222305378834189377105661966377091045314454118991031173193087912526897924892725160951529275788563515672220734346956396124580755195580737457208360330881698151150791281872981262824377757789388341314980234145096156743757137201608490271854073822699451807255412734980722522303923139699762630368196630273552286404778994356367186061509595169377391573831369366134581576852453834993464418486217647419063673896681944867898325972456231118592156982127682613245205919782212425302910167840826741063795872684191799995638895308957306891473710500615210723412961659166781986309014404583165906231102152376490342694958370772428651943018869371546229938417838030831640879222189814068049552852589928907904591512743620040242808226590755531639440474663587641583800442621697554893546303902475959432324564759328308633824603584364339546342426161944332919172661328751080766229426908974593575585594168751064122788288092970253110658025684620997932513513219016668348297372724670250463149737295309424720786806708373124828283533811649602811031549933077505719956123312473154763822358203354801083985391411380608647780315375184717889256426123881207826390856345667111087689549622297407014462564174827119086347075989403507398000446125814552260075930162803405191806571447051409515497234495906016852549900173449556022389261758330267597204479791260583751907084710023809663361150126798564574057651142063002126991201159541621775747453943960587843516042014061406344966241606812930445217144682486086239368164217096006694745966605077509824323316797350950803804890505252234540221261101717910889591913189012339997477799796213987139095979757436145623547701243809793324172469809035704392769537488590959754389766183420932857901554122280925066746908599401937569816742285545111251592218253108111831727236692103860359621084009490388877562290554929571984362562733488627462231557530889105022573563269755733906859490935411758586806725925566105583558406294236768213439681427435576356791496700020362584018145112514239866314523702612468426799243565874008251179795267820537055010513171904177107071953525134844444495085869067400340417014939117577063323464811135473991684188389774661885570535165414453111905051057117762104024480560427545927774193983919296047749377009310095034445970086509219765655517182047636586898443029903574234907473202661011203519378243032135094498673114416728146832596419034385663347357286813384988802936465607518179263898564670830743534635557764445817221560926548683524064281301373916413742068383589904430799748515969511038139397752962623760591119565128466028587643663877607033547466074995287085501250361046627570403743516803756281175571424646751675075355430905311392812384858796465341180927564728230634864747358694779915237032894025096970602371119193145800691274104728239715968069577401427649972875251086212443501572694123978428610616687291266725924118089529191570893750730572913152468572735797013293401548799051887549891469843057650306642160835888005379626809823311046391271689708582581786934263545271733175792982830583146160684634782026649400081657564243160252661053564861475565803077758704474992854774551586087323115312658624598321291437604178283274755573428740068102992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The r.h.s. exponent has $$6990$$ places meaning the value itself has more than $$10^{6989}$$ places.

• Subtraction is allowed but I don't believe making negative numbers is equivalent to subtraction as there is no zeros permitted. Commented Dec 26, 2022 at 4:38
• @TakingNotes I'd argue that -x is subtraction from an implied 0 which would appear to be perfectly legal, but ultimately OP will have to clarify. Commented Dec 26, 2022 at 4:52
• Normally I'd say the same but the OPs insistence on no other digits makes me think otherwise. Hopefully OP does get back to us on this matter though. Commented Dec 26, 2022 at 5:39
• Yes this solution using -x is allowed as explained by Albert.Lang. However I totally agree with TakingNotes, it should have been claried in the question itself. Commented Dec 26, 2022 at 18:48
• not sure whether the exponent is interesting enough to be written in full... i'd just write it as ~$10^{10^{6989.42}}$ (source) Commented Dec 27, 2022 at 7:57

I believe the largest possible number is:

115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936

obtainable by:

$$2^{4^{3+1}}$$

• Why not 31 instead of 3+1?
– RHS
Commented Dec 27, 2022 at 13:10
• The OP did not include concatenation as an allowed operation. Commented Dec 27, 2022 at 16:52

If concatenation allowed it is of the length of about 1.09795×10^19 decimal digits:

$$2^{3^{41}}$$

$$\infty$$

obtained from

$$\frac{2}{(4 - 3 - 1)}$$

• But this is not a number :P
– ACB
Commented Dec 26, 2022 at 17:28
• @ACB this is a festive answer. Commented Dec 26, 2022 at 17:39
• 2/0 is undefined, not infinity - if $a=\frac{2}{0}$, multiply both sides by 0: $0 \times a = 2$ - it cannot possibly be infinity because even for infinity, $\infty \times 0 = 0$ Commented Dec 27, 2022 at 7:49
• @somebody multiplying $\frac{2}{0}$ by $0$ gives $0$, not $2$. Multiplying anything by $0$ gives $0$. Commented Dec 27, 2022 at 8:11
• @WeatherVane right. but here we're assuming it cancels out the $0$ on the denominator, instead of being multiplied with the numerator - which should be equivalent Commented Dec 27, 2022 at 8:14