Schmittyillion

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Schmittyillion is a number coined by Trakaplex when attempting to coin the largest googolism ending in "illion".^[1] It has two different definitions coined over time, the original one in 2021, and the most recent one in 2023.

Original definition[]

The original definition of schmittyillion was introduced as "10^310^310... a googol minus 2 more times raised to the 4182001".^[1] It is intended to be a tier googol illion with respect to Bower's system, which has never been used, according to the creator. It was first clarified on Googology Testing Wiki^[2], but the defining formula included undefined convention on brackests and was based on a misconception of the behaviour of tetration.^[3] Later, it was mentioned to on a new answers.com post asking the "biggest known number in the world", but the description was ambiguous, too. Finally, the creator clarified the precise definition, as we will show in "Explanation" section. The complexity couldn't simply be written in simple notation.

Issue[]

It was unknown what the original definition "10^310^310... a googol minus 2 more times raised to the 4182001" precisely meant when the original definition was posted. Later, the creator explained it as \(10^{^{13}3\times10^{3,000}+3}\),^[3] which is still ambiguous.

Finally, the creator explained that this number is expressed in the following way:^[4]

10^{3*10^{3*10^{3*10^{3*10^{.^{.^{.^{.^{(a googol of these 3*10's)^4182001}}}}}}}(back down)}+3}

Although it includes ellipsis, it looks clearer than the original description.

More graphic Definition[]

\(10^{\underbrace{3\times 10^{.^{.^{.^{3\times 10^{4182001}}}}}}_{10^{100}\quad 3\times 10's} + 3}\) This still has ellipses, but it is easier to see what is likely meant.

Alternative definitions[]

Considering the creator's explanations, the intended value of the number is perhaps given as \(10^{a_{\textrm{googol}}+3}\) using the sequence \((a_n)_{n=1}^{\infty}\) defined in the following recursive way:

If \(n = 1\), then \(a_n := 3 \times 10^{4182001}\)
If \(n > 1\), then \(a_n := 3 \times 10^{a_{n-1}}\)

2023 Definition[]

According to a recent source, the newest definition is 10↑(3*10↑(3*10↑↑googol-1↑(4182001))+3).^[5]

Approximations[]

Notation	Approximation
Arrow notation	\(1000 \uparrow\uparrow 10 \uparrow 100\)
Down-arrow notation	\(10 \downarrow\downarrow\downarrow\downarrow 10 \downarrow 50\)
Bowers' Exploding Array Function	{1000, {10, 100}, 2}
Bird's array notation	{1000, {10, 100}, 2}
Chained arrow notation	\(1000 \rightarrow (10 \rightarrow 100) \rightarrow 2\)
Hyper-E notation	E4182001.477121255#(E100+1)
Steinhaus-Moser Notation	57[3][4]
Strong array notation	s(1000, s(10, 100), 2)
Fast-growing hierarchy	\(f_3(f_2(324))\)
Hardy hierarchy	\(H_{\omega^3 + \omega^2}(324)\)
Slow-growing hierarchy	\(g_{\varepsilon_{\omega^{\omega^2}}}(10)\)

Etymology[]

It is based off 'Schmitty Winkleson' and the word 'illion'.

See also[]

Jonathan Bowers' -illion s

Main -illions

Tier 2: millillion* · myr · micr · nan · pic · femt · att · zept · yoct · ront · quect · mec · duec · trec · tetrec · pentec · hexec · heptec · octec · ennec · icos · meicos · dueicos · trioicos · tetreicos · penteicos · hexeicos · hepteicos · octeicos · enneicos · triacont · tetracont · pentacont · hexacont · heptacont · octacont · ennacont · hect
Tier 3: killillion · meg · gig · ter · pet · ex · zett · yott · ronn · quett · hend · dok · tradak · tedak · pedak · exdak · zedak · yodak · nedak · ik · iken · icod · ictr · icter · icpet · ikect · iczet · ikyot · icron · trak · tek · pek · exak · zak · yok · nek · hot · bot · trot · tot · pot · exot · zot · yoot · not
Tier 4: kalillion · dal · tral · tal · pal · exal · zal · yal · nal · dakal · hotal · mej · dakej · hotej · gij · ast · lun · ferm · jov · sol · bet · gloc · gax · sup · vers · mult

Generalized intermediate -illions

Unofficial extensions by CompactStar and BAT4423 beyond Bowers' system

Tier 4: met/pyr · xev/gunt · hyp/kentr · omn/onl · out/paptr · barr/hous · barral · barrej · barrij · barrast · barrun · barrerm · barrov · barrol · barret · gic · asc · luc · ferc · joc · soc · bec · cet · cekal · meet · git · aset · lut · fert · jot · sot · beet
Tier 5: unvillion · duv · et · cuv · swiv · pyrv · oiv · en · ek · ev · twyv · eket · ekcuv · ekquiv · ekswiv · ekpryv · ekoiv · eknen · lan · lanunv · landuv · lanet · lancuv · lanquiv · lanswiv · lanpryv · lanoiv · lanen · etk · cuk · quik · swik · pryk · oik · enk · evaf · duvaf · etaf · cuvaf · quivaf · swivaf · pryvaf · oivaf · envaf
Tier 6: hapaxillion · dis · tris · tetrakis · pentakis · hexakis · heptakis · octakis · enneakis · decakis · hendecakis · dodecakis · decatriakis · decatetrakis · decapentakis · decahexakis · decaheptakis · decaoctakis · decaenneakis · icosakis · icosihenakis · icosidyakis · icositryakis · icositetrakis · icosipentakis · icosihexakis · icosiheptakis · icosioctakis · icosienneakis · triacontakis · tessaracontakis · pentecontakis · hexecontakis · hebdomecontakis · ogdoecontakis · enenecontakis · hecatontakis · diacosakis · triacosakis · tetracosakis · pentacosakis · hexacosakis · heptacosakis · octacosakis · enneacosakis
Generalized intermediate Tier 4: dakyr · hotyr · dakunt · hotunt · dakentr · hotentr · dakonl · hotonl · dakaptr · hotaptr · dakous · hotous · nonecxenous

Humorous extensions

Aarex Tiaokhiao's extensions in Tier 7: redillion · bittersw
Trakaplex's humorous extensions: schmitty · inphix · schmigantic (unfinished) (· helpsophiaisflirtingme (nonexistent) · schmittypleasehelpme (nonexistent)
Redstonepillager's extensions: leet
HaydenTheGoogologist2009's extensions in Tier 10: body · head · hair · face · eye · nose · ear · mouth · lip · teeth · tongu · throat · jaw · neck · shoulder · arm · elbow · hand · nail · chest · back · stomach · kne · ankle · heel · to
NigerianPidgin's extensions in Tier 10: qaq · qalp · caskisol · dicaskisol · nijicaskisol · qecask · qicask · sexicask · seqicask · ottocask · ennocask
NigerianPidgin's extensions in Tier 11: hangy · hegyek · ironikus · megmikrozz · vascor · qecask · qicask · sexicask · seqicask · gazdag · fingik · negy · rontgen · behozatal · zuhany · gyurusujj · billentyuzet · ajt · honnan · diszn · Dijillion (NigerianPidgin) · ehes · nyolcvannyolc · takar · baba · karakter · font · krim · koszonom · busz · januar · hetf · nyek · bokor · tortenelem · egyiptom · tarsashaz · tigris · pokerez · falu
XRQ CORPORATION's extensions in Tier 3: googolplex
XRQ CORPORATION's extensions in Tier 4: googolduplex
XRQ CORPORATION's extensions in Tier 5: googoltriplex
XRQ CORPORATION's extensions in Tier 6: termin · aquar · servic · spong · comb · nett · expon · adm · hecr · less · obt · playgr · check · whar · anim · grout · oreg · tront · shelf · liqu · jew · glob · ium · xiff · wemct · mibd · homr · ultim · var · squeez · tebilpictr · has · nogd · aur · trakers · icot · unth · queyn · enqu · tsun · nunt · zazt · ninj · banan · karm · guav · pizz · vist · umbrell · lav

*not Bowers'

Sources[]

↑ ^1.0 ^1.1 Trakaplex, a comment to "What is the biggest known number?" in Answers.com.
↑ Schmittyillion in Googology Testing Wiki.^{[dead link]}
↑ ^3.0 ^3.1 Trakaplex, Emphasizing Schmittyillion, Googology Wiki user blog. (The creator showed a little ambiguous expression in the blog post, and noticed a misconception in a comment.)
↑ A difference page of the talk page of this article. (The creator showed a clearer expression.)
↑ https://brettzotij.blogspot.com/2023/03/i-killillion-zillion.html (retrieved 4th March, 2023)

[root-1] 1.0 ^1.1 Trakaplex, a comment to "What is the biggest known number?" in Answers.com.

[2] Schmittyillion in Googology Testing Wiki.^{[dead link]}

[explanation-3] 3.0 ^3.1 Trakaplex, Emphasizing Schmittyillion, Googology Wiki user blog. (The creator showed a little ambiguous expression in the blog post, and noticed a misconception in a comment.)

[4] A difference page of the talk page of this article. (The creator showed a clearer expression.)

[5] ttps://brettzotij.blogspot.com/2023/03/i-killillion-zillion.html (retrieved 4th March, 2023)

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