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Flan number

Flan numbers (Japanese: フラン数) are numbers coined by Japanese googologist 巨大数大好きbot. There are several versions of Flan numbers.


The numbers are named after Flandre Scarlet (Japanese: フランドール・スカーレット, nickname: フラン)[1], who is a famous character in Japanese game series 東方Project.

One of the most famous versions Flan number 4 version 3 (Japanese: フラン数第四形態改三) was coined on 03/22/2018.[2] Its predeccessor Flan number 4 version 2 (Japanese: フラン数第四形態改二) is known to be equal to 6, and is the origin of the tradition in one of Japanese googology communities that \(6\) is a large number.[3][4][5]

Perhaps this episode is the origin of the number later coined by Japanese googologist Nayuta Ito.


The following is a table of informations of versions.[6][7]

version name approximation
フラン数第一形態 \(10^{10^{10^{670767}}}\)
フラン数第二形態 \(f_{\omega^{\omega^{\omega}}}^6(5)\)
フラン数第三形態 \(f_{\varepsilon_0}(6)\)
フラン数第四形態 ill-defined
フラン数第四形態改 ill-defined
フラン数第四形態改二 \(6\)
フラン数第四形態改三 \(f_{\varepsilon_4^{\varepsilon_4^{\varepsilon_4}}}(5)\)
フラン数第五形態改二 \(5\)
フラン数第五形態改三 \(f_{\varepsilon_{\omega^{\omega^{\omega^{\omega+2}}}}}(5)\)
フラン数第六形態改二 ill-defined
フラン数第六形態改二甲 ill-defined
フラン数第七形態改二 ill-defined
フラン数準第八形態改二 ill-defined
フラン数最終形態・冬 \(f_{\varepsilon_{\omega^{\omega^{\omega^{\omega}}}}}(5)\)
フラン数最終形態・終 canceled
フラン数最終形態・了 ill-defined
フラン数最終形態・焉 ill-defined
フラン数最終形態・完 ill-defined

Here, \(f\) denotes the fast-growing hierarchy associated to Veblen hierarchy.

Japanese letters

The word "フラン" means "Flan", i.e. the nickname of Flandre Scarlet, and the letter "数" means "number". Therefore the combination "フラン数" means "Flan number". The letter "第" means "-th", and the letters "一", "二", "三", "四", "五", "六", "七", and "八" mean "1", "2", "3", "4", "5", "6", "7", and "8" respectively. Therefore for example, the combination "第三" means "third". The word "形態" means "form" or "version". The letter "改" means "revision", and hence the combination "改二" means "revision 2", i.e. the second revision. The letter "甲" means "first" in one of Japanese old counting systems. Therefore the combination "フラン数第六形態改二甲" means "The first version of the second revision of the sixth form of Flan number".

The word "形態" means "form" or "shape", and "最終" means "final". Therefore "フラン数最終形態" means "The final form of Flan number", although there are five forms of フラン数最終形態. The word "冬" for the first final form means the winter, but is just regarded as the right half side of the word "終". The word "終" for the second final form means "finished", and hence "フラン数最終形態・終" means "The final form of Flan number - the finale". The word "了" for the third final form means "completed", and hence "フラン数最終形態・了" means "The final form of Flan number - the denouement". The word "焉" for the fourth final form means "terminated", and hence "フラン数最終形態・焉" means "The final form of Flan number - the terminus". The word "完" for the last final form means "concluded", and hence "フラン数最終形態・完" means "The final form of Flan number - the end".

External links


See also

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's \(\psi\)
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 computation programmes · TR function (I0 function)
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence computation programme · ω-Y sequence computation programme
By Nayuta Ito: N primitive · Flan numbers · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4
By p進大好きbot: Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence · YY sequence · Y function · ω-Y sequence
See also: Template:Googology in Asia