Flan numbers

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.
== Episode ==
The numbers are named after Flandre Scarlet (Japanese: フランドール・スカーレット, nickname: フラン)ピクシブ百科事典 - フランドール・スカーレット, 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.巨大数大好きbot, Definition of Flan number 4 version 3, twitter. Its predecessor 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 with the Japanese phrase "6は巨大数", partially as a sort of a joke.叢武, 6は巨大数, twitter.巨大数大好きbot, Definition of Flan number 4 version 2, twitter.rpakr, The statement that Flan number 4 version 2 coincides with 6, twitter.Hexirp, What is the origin of 6?, Japanese Googology Wiki user blog.
Perhaps this episode is the origin of the number ６ later coined by Japanese googologist Nayuta Ito.
== Versions ==
The following is a table of informations of versions.FHLASR, 東方巨大数2エントリー一覧, Google spreadsheet.FHLASR, 第4回東方巨大数巨大数投稿一覧, Googleスプレッドシート.
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 	\(f_{\varepsilon_0}^7(6)\)
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 ==
* 西宮 七南-解説channel, ゆっくりと学ぶ巨大数論～フラン数第一形態～, YouTube. (An introduction video of フラン数第一形態)
* 西宮 七南-解説channel, ゆっくりと学ぶ巨大数論～フラン数第二形態～, YouTube. (An introduction video of フラン数第二形態)
* 猫山にゃん太, Flan Number 4th - フラン数第四形態 改二/改三, GitHub Page. （A javascript code to compute expansions of フラン数第四形態改二/改三）
== Sources ==
== See also ==
* :ja:フラン数第一形態
* :ja:フラン数第二形態
* :ja:フラン数第四形態改三
ja:フラン数
zh:芙蘭數
Category:Numbers
Category:Googology in Asia
Category:Incomplete