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The inphixillion is a number introduced as tier googolillion illion[1]. The term was coined by Trakaplex. Just like schmittyillion, it was first clarified on Googology Testing Wiki[2] without a precise definition, and was later described on an answers.com post as "in tier googolillion (10^(3*googol+3))".[1] There has been rumors that the creator has been working on an illion disastrously larger, but no word has slipped yet.

## Issue

The number was introduced as 10(3*10↑↑103*10100+3)110,101,011(↓↓103*10100+3-1)+3, but the unary operation ↓↓ is undefined.[3] Since ↓↓ itself is not defined on numbers, one needs to precisely formalise a notation including ↑↑ and ↓↓. Moreover, the creator tends to write (3*10↑↑b)c even if it is not intended to be the c-th power of 3×(10↑↑b) or 3 times the c-th power of 10↑↑b. It perhaps means a value of the sequence $$a_n$$ introduced in "Alternative definition" section.

According to the creator, the definition means "a googolillion 3*10 base exponents raised to the 110101011 back down a googolillion bases plus three", which is still ambiguous. Perhaps it can be described in a way similar to schmittyillion#Alternative definition, as we will explain in "Alternative definition" section.

## Alternative definition

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

1. If $$n = 1$$, then $$a_n := 3 \times 10^{110101011}$$
2. If $$n > 1$$, then $$a_n := 3 \times 10^{a_{n-1}}$$

Readers should be careful that $$a_n$$ is not something like $$3 \times (10 \uparrow \uparrow n)^{110101011}$$ or $$((3 \times 10) \uparrow \uparrow n)^{110101011}$$, although the creator used the expression (3*10↑↑103*10100+3)110,101,011. According to the creator, this alterntive definition is compatible with the creator's intention.[4]

More generally, define a map \begin{eqnarray*} f \colon \mathbb{N}^2 & \to & \mathbb{N} \\ (x,y) & \mapsto & f(x,y) \end{eqnarray*} in the following recursive way:

1. If $$y = 0$$, then $$f(x,y) := x$$
2. If $$y > 0$$, then $$f(x,y) := 3 \times 10^{f(x,y-1)}$$.

Then the alternative definition of schmittyillion is equivalent to $$10^{f(4182001,\textrm{googol})+3}$$, and the alternative definition of inphixillion is equivalent to $$10^{f(110101011,\textrm{googolillion})+3}$$.

## Etymology

It is based off 'inphixinity' and the word 'illion'.

## Sources (in progress)

1. what is the biggest known number Answers.com. 1/18/2021.
2. Inphixillion. Googology Testing Wiki. 1/18/2021.
3. Trakaplex, A comment explaining ↓↓ but clarifying the lack of the definition, Googology Wiki user blog.