Power Towers

In this article, we will cover "power towers".

Double Exponential Numbers

We will begin with the double exponential numbers. A double exponential number is a number of the form a^(a^b) (where a and b are integers), and are the three-height power tower numbers. Robert Munafo refers to these numbers as powerlogs.

We will begin with the base-2 double exponential numbers.

2^2^1 = 4

2^2^2 = 16

2^2^3 = 28 = 256

2^2^4 = 216 = 65,536

2^2^4 is over 65 thousand, and so is quite sizable in terms of everyday numbers.

2^2^5 = 232 = 4,294,967,296

2^2^5 is already about 4.3 billion. This is approximately 94% of the number of years since the Solar System came into existence, and this many years going into the past would be about 31% of the history of the universe.

2^2^6 = 264 = 18,446,744,073,709,551,616

The sixth double exponential of 2 is greater than the number of millimeters in one light-year! 

2^2^7 = 2128 = 340,282,366,920,938,463,463,374,607,431,768,211,456

The seventh double exponential of 2 is a number of 39 digits. This many picometers would be nearly the entire way across the observable universe. This number is also comparable to the number of years from now by which there will be 1/17179869184 of the amount of particles there are in the universe today if the proton is unstable and has a half-life of 1037 years.

Now we will try base 3.

3^3^1 = 33 = 27

3^3^2 = 39 = 19,683

3^3^3 = 327 = 7,625,597,484,987

3^3^4 = 381 = 443,426,488,243,037,769,948,249,630,619,149,892,803

3^3^4 is about as big (and actually a bit larger than) 2^2^7.

3^3^5 = 3243 = 87,189,642,485,960,958,202,911,070,585,860,771,696,964,072,404,731,750,085,525,219,437,990,967,093,723,439,943,475,549,906,831,683,116,791,055,225,665,627

3^3^5 is already about 8.72 quintillion times a googol. Compare this with 2^2^8, which is still less than the reciprocal of 8.7*1015 (or about 1.14*10-16) times a googol.

Now, we will cover double exponentials of 10.

10^10^1 = 1010 = 10,000,000,000

Even the first double exponential of 10 is already equal to ten billion. This many years is about 72.5% of the time the universe has existed, and even just this many seconds is close to 200 years longer than the longest time anyone has ever lived.

Triple Exponential Numbers

Next up we have triple exponential numbers. These are numbers of the form a^(a^(a^b)), and as you will see, this is a HUGE improvement over the double exponential. First, we will look at triple exponentials of 2.

2^2^2^1 = 2^2^2 = 24 = 16

2^2^2^2 = 2^2^4 = 216 = 65,536

2^2^2^3 = 2^2^8 = 2256 = 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

Even just 2^2^2^3 is already a number of 78 digits! And the next one is 2^2^2^4, which is 2^2^16 or 265536 (which is the same as 2^^5 (for more, see the next article), a number of 19,729 digits beginning with 200352993040..., and ending with ...905719156736. The next triple exponential after that, 2^2^2^5, is a number with over a billion digits beginning with 3103285..., and ending with ...4691982336.

Below are the first and last digits of more triple exponentials of 2.

65536

2^2^2^3: 115792089237316195423570985008687907853269984665640564039457584007913129639936

2^2^2^4: 20035299304068464649790723515602557504478...55822087777506072339445587895905719156736 (19,729 digits)

2^2^2^5: 31032805438632861402998911558636455910328...458793500608595555903271649464691982336 (1,292,913,987 digits)

2^2^2^6: 19069740116044733845522417467451879839283...367004310804367570259007866971447361536 (5,553,023,288,523,357,133 digits)

2^2^2^7: 11899811685296656534306723819955291715435...514316717645972216153903192091641511936 (102,435,199,438,739,363,750,012,109,250,103,232,701 digits)

2^2^2^8: 37925828077823206940504769042642762359569...528072187720249599620404862789391220736 (over 3.48*1076 digits)

2^2^2^9: 15365619647315663399281284825759039147932...4383957186718990336 (over 4.03*10153 digits)

Now we will cover triple exponentials of 3.

3^3^3^1 = 3^3^3 = 327 = 7,625,597,484,987

3^3^3^1 is equal to about 7.6 trillion, which is now nothing surprising. But the very next one is:

3^3^3^2 = 3^3^9 = 319683

The result of 3^3^3^2 is a number with 9,392 digits! Its complete decimal expansion is below.

15054164145220926243143298033398654543076143473537427823628361588037621543571301695623049264407339299686835128946425225443147826949787930561330768130940971342651498914747987903017936839366686681222858853212428390499179112260835999559081805771312604025295439082557215698027337358383758012148589502083748215403098007878995911835889729355019007999648624818474775244389973945623504736442334340264449180923862032373721103041062506593780125745533233578404272817694355494119034087892415581159553444177665845561067414178027984259354275578627111015392093577703217842246757715993850527863143440137825302954400096599965422035350068885003919811126843083535495234137589102455390614147745453121569003468531097431528705505762776585634579969292789571943150663200583121057376633971559508145755565137282097601570923832022078513418953646013882192412910190940854648052228760154965309445367023936846368878240702513600430947255850771152764992919320215061847218258666628853530235012194712957400384839776451713368743915557078445061496138530357561367254529501546194141026743692552770867530443497697863635747556324459847110054346253217411309820943763589568070626933220130999528465911723696379451234379509178882111255496057359862889587710804096797256376246508804835264315162962684856485371330030695400531740336618389165874799967876888281848913327759470167768279402682276707870081224732274384659249880078341650469661298082124051010731079314524349989388847576030971007806934004658380045876439915172218071407398096425454863499964175936045588899511100818127296423649675861180575299409125634578831950500582196834881847467913599246402743804938245328780371835700553894646696439253532812485180678663530963320404345804119023110188753228948399337925324541338866774767148916824570033330260935113206834204238946077142955428600706271826005072958108100916095852989600626381105886901811561810802820260654105888681595530534070524243859292040849898489759577425880929604186454171395384746747160467644222788246631818591794851263662282260043766820614615742247946680888917609431111964443845450113012016059028252949251444370511016864338540421007357077318071754263259398776843989268150427952995598033652815480803298792193004600028845636323440783769489575586689253910521580631176759695926429858708622527151950398108778525193593810341416763770438316694501847590075040210810716906685947380077182469744725823543447706588324768834763195562774985477798540217873298009604291129795418962227074102940207034948505153392457242087232988694897916918627938886485340394230939699631804266400986148520789113998942104474638515790497298177180948117347668881428380897994432629860113081048070903346286687397585575637777736565140091811959181405390636748958073464466557363547202664370524151400189210741771647667960261006847774366070395219311527186379209439027226907110564971290375618800743037163479167095483637962017820862188917071553305789955126588684827797726211120735968197477850571316770150287165878389839672342545545460165773088511587169977051585969513304977545948135123659981533754846412899129174484606253824123635632075542515296604150671865907958836281902907426790237882271302563288613436612237649386048826478270866889971585530976643980567559081588367994216316415534377310625210842978347522749998304053196752201754291357595866691031427438570488413436982159007252192702526115435650456773187187481565525939418031338645381854899072443064229271140086252730267748967711958180498611607049993203914690982677948315892617601093090131230276263165247404303103302830312349238931851942362810611095371728245035650497938755128164297995830988798545113694176618213622919275401110086443810298346987303592255976821255980532657116421770907895216347078843396127764921952923456274228566429587332874013896763727661069379600730502195223315312967115751770637900246922470093827182690659231910787467937864884211138887041168975221581615809725014624250894939079035701448869701225935495734868690083108274643511778596506264788318426355551685325057973634424815602788020887050644386589470775779265347762598603818475791916367797747391246095655602223800648036700788484402544656890674685922345400047918107273508338752592743072335213702513608098172675113664693724883548584507464227538749558370794862387028827830419122950052575201685936407713769282830574759622625902354074518375724965919525401514696438912583055540778987018405913542432995957670724845442314460803725488542761282286616862144421242539811410033272661429024136968270536203686998930251635251487644468713748016989077058209715435551766851283841007354515875080796133195350862867811292109287044222663016132495315942566169106389856554513738407490597282746783788129812979028109503241161061784577387960452863724350171347066771273284045313314470700496181528490374495712531002858110305051595394688910479451764010214199411507762270707265034786083018480608796302425087147470519889489244386430664752210390362287006968230507058193139036852298914671416942087177576681090814377516707134370882317576008149318881936452031196608278334775755046577502950933021658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And if you thought that was amazing, just look at this:

3^3^3^3 = 3^3^27 = 37,625,597,484,987

The result of 3^3^3^3 is a number with over three trillion digits! The first and last digits of this number are 12580142906... ...776100739387.

Now, we will cover triple exponentials of 10.

10^10^10^1 = 10^10^10 = 1010000000000

10^10^10 is equal to 10 to the power of 10,000,000,000, or one followed by 10 billion zeroes. But then the next one is:

10^10^10^2 = 10^10^100

The second triple exponential of 10 is exactly equal to a googolplex.

Quadruple Exponential Numbers

We will continue by covering quadruple exponential numbers, numbers of the form a^(a^(a^(a^b))). There are only a few such numbers that we can even know the first leading digit of. Of course, we will begin with base 2.

2^2^2^2^1 = 2^2^2^2 = 2^2^4 = 216 = 65536

2^2^2^2^2 = 2^2^2^4 = 2^2^16 = 265536 = 200352993040684646497907235156025575...587895905719156736 (19729 digits)

2^2^2^2^3 = 2^2^2^8 = 2^2^256 = 2115,792,089,237,316,...,007,913,129,639,936 (78 digits) = 37925828077823206940...620404862789391220736 (approximately 3.485689*1077 digits)

Even just 2^2^2^2^3 is in the vicinity of the upper-bound on the number of years until all matter will collapse into black holes if the proton is stable, and in fact the next one, 2^2^2^2^4, has over 1019727 digits beginning with 212003872880821..., and ending with ...277507437428736.

And, 2^2^2^2^5 has over 9.34*101292913985 digits beginning with 315921269337233843004184822..., and ending with ...8478717272631317364736. Below are the first and last 100 digits:

3159212693372338430041848223251106402238836591165032496711917068918329936709841286666242144604280394...

...8615067995358861563348858582700539981272322967957571470169579936741967543549668478717272631317364736

In fact, this is the largest number I have directly computed the first digits of (although it is still unknown whether Mathematica is actually reliable with calculations of this size).

The first quadruple exponential of 3 is 3^3^3^3^1 or 37625597484987. The next one is:

3^3^3^3^2 = 3^3^3^9 = 3^3^19683 = 315054164145...86617859227 (9392 digits)

The second quadruple exponential of 3 has over 7.1866*109390 digits! This number is not only much larger than a googolplex, but it is also larger than the googolplexibell, or 10^10^5000. Below are the first and last 100 digits of this number.

1024072831893982752343129633195300517870598945274774443978826667764518244565368173937762661544334026...

...4295945033360670777334434803856013811295833893351707423776655910723048491894212709956633607528172987

And the third quadruple exponential of 3, 3^3^3^3^3, is equal to 3^3^7625597484987, or 3 to the power of a number with over 3.638 trillion digits!