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 A specific value of the Ackermann function tetrator Junior Fellow Posts: 3 Threads: 1 Joined: Oct 2008 10/16/2008, 11:43 PM Could someone who has the capability please compute Ack(61,61) for me? Thanks in advance bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 10/17/2008, 11:50 AM tetrator Wrote:Could someone who has the capability please compute Ack(61,61) for me? Thanks in advanceDid you think before you wrote? Already $\text{Ack}(4,4)=2 ^ {2 ^{2^{2^{2^{2^{2}}}}}-3 =2^{2^{2^{65536}}}}-3$ as you can see in the table of values in the wikipedia article about the Ackermann function. As a pretaste of how big the numbers become, here is only $2^{65536}$: 20035299304068464649790723515602557504478254755697514192650169737108940595563114530895061308809333481010382343429072631818229493821188126688695063647615470291650418719163515879663472194429309279820843091048559905701593189596395248633723672030029169695921561087649488892540908059114570376752085002066715637023661263597471448071117748158809141357427209671901518362825606180914588526998261414250301233911082736038437678764490432059603791244909057075603140350761625624760318637931264847037437829549756137709816046144133086921181024859591523801953310302921628001605686701056516467505680387415294638422448452925373614425336143737290883037946012747249584148649159306472520151556939226281806916507963810641322753072671439981585088112926289011342377827055674210800700652839633221550778312142885516755540733451072131124273995629827197691500548839052238043570458481979563931578535100189920000241419637068135598404640394721940160695176901561197269823378900176415171900511334663068981402193834814354263873065395529696913880241581618595611006403621197961018595348027871672001226046424923851113934004643516238675670787452594646709038865477434832178970127644555294090920219595857516229733335761595523948852975799540284719435299135437637059869289137571537400019863943324648900525431066296691652434191746913896324765602894151997754777031380647813423095961909606545913008901888875880847336259560654448885014473357060588170901621084997145295683440619796905654698136311620535793697914032363284962330464210661362002201757878518574091620504897117818204001872829399434461862243280098373237649318147898481194527130074402207656809103762039992034920239066262644919091679854615157788390603977207592793788522412943010174580868622633692847258514030396155585643303854506886522131148136384083847782637904596071868767285097634712719888906804782432303947186505256609781507298611414303058169279249714091610594171853522758875044775922183011587807019755357222414000195481020056617735897814995323252085897534635470077866904064290167638081617405504051176700936732028045493390279924918673065399316407204922384748152806191669009338057321208163507076343516698696250209690231628593500718741905791612415368975148082619048479465717366010058924766554458408383347905441448176842553272073155863493476051374197795251903650321980201087647383686825310251833775339088614261848003740080822381040764688784716475529453269476617004244610633112380211345886945322001165640763270230742924260515828110703870183453245676356259514300320374327407808790562836634069650308442258559670392718694611585137933864756997485686700798239606043934788508616492603049450617434123658283521448067266768418070837548622114082365798029612000274413244384324023312574035450193524287764308802328508558860899627744581646808578751158070147437638679769550499916439982843572904153781434388473034842619033888414940313661398542576355771053355802066221855770600825512888933322264362819848386132395706761914096385338323743437588308592337222846442879962456054769324289984326526773783731732880632107532112386806046747084280511664887090847702912081611049125555983223662448685566514026846412096949825905655192161881043412268389962830716548685255369148502995396755039549383718534059000961874894739928804324963731657538036735867101757839948184717984982469480605320819960661834340124760966395197780214411997525467040806084993441782562850927265237098986515394621930046073645079262129759176982938923670151709920915315678144397912484757062378046000099182933213068805700465914583872080880168874458355579262584651247630871485663135289341661174906175266714926721761283308452739364692445828925713888778390563004824837998396920292222154861459023734782226825216399574408017271441461795592261750838890200741699262383002822862492841826712434057514241885699942723316069987129868827718206172144531425749440150661394631691976291815065797455262361912248480638900336690743659892263495641146655030629659601997206362026035219177767406687774635493753188995878662821254697971020657472327213729181446666594218720034745089428309115351892711142871083761592223802766053278233516615551493693757784666701457179719012271178127804502400263847587883393968179629506907988171216906869295382485298300234760684541141781391106485602365497542274972310076151318700240539105109138178437217914225285874320985249578780346837033378184214440171386881242499844186181292711985333153825673218704215306311977485352146709553346263366108646673322924098798492566911095161436186015489097402419135096230436121961281659505186660220307156136847323646608689050142639139065150639081993788523183650598972991254044794434251667742996598118492331515552728832740283526884424087528112832899806259126736995462473415433335001472314306127503903073971352520693381738433229507010490618675394331307847980156551303847581556852362180104196502555961819349863159132330360964619059902361126811960234418433633345949276319461017166529138237171823942992162725384617760656945422978770713831988170369645886898118632109769003557358846244648357062914530527571012788720279653644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Finitist Junior Fellow Posts: 5 Threads: 0 Joined: Sep 2008 10/19/2008, 11:29 AM (This post was last modified: 10/19/2008, 11:40 AM by Finitist.) I think this is a board to be read slowly and with pen and paper to hand (unless you're a mathematics genius, which I'm not). As a first observation, it's the sheer size of the numbers obtained by tetration and other hyperoperators that attracts me to the topic. bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 10/19/2008, 11:56 AM Finitist Wrote:I think this is a board to be read slowly and with pen and paper to hand (unless you're a mathematics genius, which I'm not).At least I expect that one is willing to deal with the question. As opposed to an "serve me" attitude. Quote:As a first observation, it's the sheer size of the numbers obtained by tetration and other hyperoperators that attracts me to the topic. Then you probably already read, what Robert Munafo has to say about the topic: http://www.mrob.com/pub/math/largenum.html martin Junior Fellow Posts: 14 Threads: 1 Joined: Jul 2008 10/20/2008, 07:13 PM While we're on the topic of large numbers, has anyone ever computed the last few digits of the Moser? I only know it ends in ...56. Is it correct that 2(triangle) = "Zelda" ends in ...42656? Not that I really need to know it, I'm just curious. @ tetrator: Why do you specifically want to know that number? Any background story? Thanks. tetrator Junior Fellow Posts: 3 Threads: 1 Joined: Oct 2008 10/20/2008, 11:49 PM Thanks for asking, Martin. I'm working with a function from N to N whose value at 1 is 2, its value at 2 is 7, and its value at 3 is A(A(A(61,61),A(61,61)), A(A(61,61),A(61,61))). I know A(61,61) is somewhat large, but I thought maybe someone with enough ram could pound it out so it would be more clear that (A(61,61), A(61,61)) is just a 2-tuple and so makes a valid argument for the next A in the nest f(4) is A(A(A(A(A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))))), A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))))), A(A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))))), A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))))))), A(A(A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))))), A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))))), A(A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))))), A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))))))), A(A(A(A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))))), A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))))), A(A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))))), A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))))))), A(A(A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))))), A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))))), A(A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))))), A(A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4)))), A(A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))), A(A(A(4,4),A(4,4)), A(A(4,4),A(4,4))))))))) martin Junior Fellow Posts: 14 Threads: 1 Joined: Jul 2008 10/21/2008, 03:27 PM (This post was last modified: 10/21/2008, 03:28 PM by martin.) Oy. This amount in dollars would sure help the world out of its financial crisis. bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 10/21/2008, 07:14 PM (This post was last modified: 10/21/2008, 07:15 PM by bo198214.) martin Wrote:Oy. This amount in dollars would sure help the world out of its financial crisis. Unfortunately there can never be so many dollars, because the number is bigger than the number of atoms in the universe. Already the number $2^{65536}$ which I gave in my initial post, which is much much much much (its really not expressable how incredibly much) smaller than the number tetrator was seeking is around $10^{19728}$, while the number of atoms in the universe (see wikipedia) is around $10^{80}$. martin Junior Fellow Posts: 14 Threads: 1 Joined: Jul 2008 10/21/2008, 08:13 PM bo198214 Wrote:martin Wrote:Oy. This amount in dollars would sure help the world out of its financial crisis. Unfortunately there can never be so many dollars, because the number is bigger than the number of atoms in the universe. An amusing comparison. True, but amusing. By the way, I wonder how Arthur Eddington came up with 15747724136275002575605653961181555468044717914527116709366231425076185631031296 as the number of protons in the universe. At least close to 10^80 he was. Finitist Junior Fellow Posts: 5 Threads: 0 Joined: Sep 2008 10/26/2008, 06:23 PM (This post was last modified: 10/26/2008, 06:46 PM by Finitist.) tetrator Wrote:Thanks for asking, Martin. I'm working with a function from N to N whose value at 1 is 2, its value at 2 is 7, and its value at 3 is A(A(A(61,61),A(61,61)), A(A(61,61),A(61,61))). Hi. I've seen a sequence like that on Robert Munafo's site. Is yours related to Friedman sequences (nonrepeating sequences of different numbers of letters) as described below? Here's the extract from the relevant page; Friedman Sequences In a 1998 paper, Harvey Friedman describes the problem of finding the longest sequence of letters (where there are N allowed letters) such that no subsequence of letters i through 2i occurs anywhere further on in the sequence. For 1 letter the maximum length is 3: AAA. For 2 letters the longest sequence is 11: ABBBAAAAAAA. For 3 letters the longest sequence is very very long, but not infinite. He then goes on to show how to construct proofs of lower bounds for N-character sequences using certain special (N+1)-character sequences. With help from R. Dougherty, he found a lower bound for the N=3 case, A7198(158386) = ack-rm(7198,158386) = ack(7198,2,158386) = hy(2,7199,158386) = 2(7199)158386, where x(7199)y represents the 7199th hyper operator. http://www.mrob.com/pub/math/largenum-4.html It's amazing that a function can jump suddenly like that from mundane numbers to stratospherically huge ones, in fact I think it qualifies as a mathematical *joke. I'd be interested to know what that function is that you're working on, unless you'd rather keep it private. * Apologies if this is offtopic but I find the expression for the "Look and Say" sequence funny too, or rather the fact that a sequence that's so simple to express needs such a complicated polynomial to express it; http://en.wikipedia.org/wiki/Look_and_say_sequence « Next Oldest | Next Newest »

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