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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)[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.

## Versions

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".