a< back to index 𝜑 Function Family

1. Introduction[Under Construction] 2. The 𝜑 Function (Regular 𝜑 Function)We want to have certain function that iterates the number of factorial-like notation. after the number n. For example (using regular factorial):3!!…!!⏠⏣⏣⏡⏣⏣⏢100That number is big, surely, but difficult to write. So, the idea is we will using some function that do the number of "trailing factorial-like notation" after the argument (n). In this example, we use !h to represent any factorial-like notation that we've discussed in part 1. (h stands for hyper-factorial). Trivia: 𝜑 in greeks alphabet is "phi", and the pronounciation is close enough to f, which is, "phactorial". 2.1. Zeroth 𝜑 FunctionDefinition and notation

	𝜑00(n)	=n
	𝜑10(n)	=n!h
	𝜑20(n)	=n!h!h
	𝜑30(n)	=n!h!h!h
	⋮	= ⋮
	𝜑m0(n)	=n!h ...!h⏠⏣⏣⏡⏣⏣⏢m

The definition above is the "zeroth 𝜑" function. It just count how many factorial-like notation after the n. Example:

	𝜑10(3)	=3!h
	𝜑40(3)	=3!h…!h⏠⏣⏣⏡⏣⏣⏢4=3!h!h!h!h

2.2. First 𝜑 Function (𝜑1)

	𝜑01(n)	=𝜑00(n)=n
	𝜑11(n)	=𝜑n0(n)

The first iteration of 𝜑1 function (𝜑11) is just 𝜑0 with n-many hyper factorial (n iteration). But, what if the number of hyper factorial itself is 𝜑0 with n-many hyper factorial? That is, my friends, the second iteration of 𝜑1 function (𝜑21). Here we have the example for second and third iteration of 𝜑1 function.

	𝜑21(n)	=𝜑𝜑n0(n)0(n)
	𝜑31(n)	=𝜑𝜑n00𝜑n0(n)(n)

How's that works? Well, let's unpack.

	𝜑21(n)	=𝜑𝜑n0(n)0(n)
		=n!h…!h⏠⏣⏣⏡⏣⏣⏢𝜑n0(n) times
	𝜑31(n)	=𝜑𝜑n00𝜑n0(n)(n)
		=n!h…!h⏠⏣⏣⏡⏣⏣⏢n!h…!h⏠⏣⏣⏣⏡⏣⏣⏣⏢𝜑n0(n)

Alright, you get the idea. The next question is, can we compress the notation 𝜑1 function to the simpler one? Okay, it may be still easy to write when the iteration is only 2 or 3. But, what about 𝜑9991. That's too many tower of 𝜑n0 stacked on top each other. If you notice, 𝜑21(n) and 𝜑31(n) can be defined recursively as following:

	𝜑21(n)	=𝜑𝜑11(n)0(n)
	𝜑31(n)	=𝜑𝜑21(n)0(n)

Henceforth, we can write the generalized notation of the first 𝜑 function𝜑m1(n)=𝜑𝜑m-11(n)0(n)Example: If m=1𝜑11(n)=𝜑𝜑1-11(n)0(n)=𝜑𝜑01(n)0(n) We know that 𝜑01(n)=𝜑00(n)=n. Thus,𝜑11(n)=𝜑𝜑01(n)0(n)=𝜑n0(n) Which is consistent with our system. 2.3. Second 𝜑 Function

	𝜑02(n)	=𝜑01(n)=𝜑00=n
	𝜑12(n)	=𝜑n1(n)

But, what if the iteration (The superscript m in 𝜑m1(n)) itself is at the level of 𝜑1 with n-many iteration? You can guess it ... yup. It is the second iteration of 𝜑2 function (𝜑22).

	𝜑22(n)	=𝜑𝜑n1(n)1(n)(1)
		=𝜑𝜑12(n)1(n)

And for the third iteration:

	𝜑32(n)	=𝜑𝜑11𝜑21(n)(n)
		=𝜑𝜑22(n)2(n)

And so on. We can write the generalized notation of the second 𝜑 function.𝜑m2(n)=𝜑𝜑m-12(n)1(n)You can try plugging m=1 and see the same consistent results (namely 𝜑n1(n)). 2.4. wth 𝜑 FunctionGeneralized notation of the wth 𝜑 function (wth here denotes nth 𝜑 function. But since n has been taken by the argument of 𝜑 function, we use w).𝜑mw(n)=𝜑𝜑m-1w(n)w-1(n)Here, we calls w as "the level of 𝜑 function", m as "the iteration", and n inside the function as "the argument". Is the generalization correct? Let's try plugging some number. If w=2 and m=2

	𝜑22(n)	=𝜑𝜑2-12(n)2-1(n)
		=𝜑𝜑12(n)1(n)
		=𝜑𝜑n1(n)1(n)

Which is correct (see (2)) 2.5. Beyond nth 𝜑 Function: Next RecursionWhat if the level of function (the w in 𝜑mw(n)) itself is at the level of 𝜑0 with n-many hyper factorial? We will dive in the next function, stronger function, the moderate 𝜑 function. We will write it as ϕ(n). 3. ϕ(n) Function (Moderate 𝜑 Function)We want to have certain function that iterate the level of function 𝜑mw(n) at the strength of 𝜑(n). 3.1. First ϕ(n) Function (ϕ1(n))The idea is, we want to have notation that takes the level of 𝜑 function (the w in 𝜑mw(n)) to the strength of 𝜑(n) itself. Definition and notationϕ1(n)=𝜑n𝜑nn(n)(n) The first level ϕ(n) defined as 𝜑(n) with the level of 𝜑(n) equal to 𝜑n strength. Take a look that ϕ(n) doesn't have iteration (the "superscript"), only level (the "subscript"). Instead, it automatically made the iteration of 𝜑 function (the m in 𝜑mw(n)) equal to n. 3.2. Second ϕ(n) Function (ϕ2(n))Definition and notation

	ϕ2(n)	=𝜑n𝜑n𝜑nn(n)(n)(n)
		=𝜑nϕ1(n)(n)

3.3. wth ϕ(n) Function (ϕw(n))Definition and notation

ϕw(n)

=𝜑nϕw-1(n)(n)

4. Φ(n) Function (Strong 𝜑 Function)4.1. Φ𝛼(n)The zeroth Φ𝛼 (Φ𝛼0) is the same with n-level ϕ function, that is ϕn(n).Φ𝛼0(n)=ϕn(n) The first Φ𝛼 (Φ𝛼1) is ϕ function with the hyper-level of ϕ(n), n times. What? Take a look.

Φ𝛼1(n)

=ϕϕ⋱⋰(n)⏠⏣⏣⏣⏡⏣⏣⏣⏢n(n)

It is easier to understand with recursive definition. Let an be the hyper-level of ϕ(n), n-times.

	an	=ϕan-1(n)
	an-1	=ϕan-2(n)
	⋮	⋮
	a2	=ϕa1(n)
	a1	=ϕa0(n)=ϕn(n)

Thus, Φ𝛼1(n) defined recursively as following

	Φ𝛼1(n)	=ϕan(n)
		=ϕϕ⋱⋰(n)⏠⏣⏣⏣⏡⏣⏣⏣⏢n(n)

This is absolute humongous, since Φ𝛼1 not only iterates the number of level n-times, but it is recursively iterates the number of level of level of level of .... level of⏠⏣⏣⏣⏣⏣⏣⏣⏡⏣⏣⏣⏣⏣⏣⏣⏢n-times (ϕnn)-times. You get the idea. It is absolute humongous and we still in the first level of Φ𝛼. The second Φ𝛼 (Φ𝛼2) is ϕ function with the hyper-hyper-level of ϕ(n), n times. Take a look.

	Φ𝛼1(n)	=ϕϕ⋱⋰(n)⏠⏣⏣⏣⏡⏣⏣⏣⏢ϕ⋱⋰(n)⏠⏣⏣⏣⏡⏣⏣⏣⏢n(n)
		=ϕaan(n)

The level of ϕ function in the Φ𝛼2 function is aan. This is called hyper-hyper-level, or hyper2 level. Therefore, we can define Φ𝛼w(n) for any w-level as hyperw level of ϕ function.Recursively,

	Φ𝛼w(n)	=Φ𝛼(w-1)(n)
		=ϕa⋱⏦w-times(n)

4.2. Φ𝛽(n)4.2.1. Φ𝛽0The zeroth Φ𝛽 (Φ𝛽0) is the same with n-level Φ𝛼 function, that is Φ𝛼n.Φ𝛽0(n)=Φ𝛼n(n)4.2.2. Φ𝛽1The first Φ𝛽 (Φ𝛽1) is defined

Φ𝛽1(n)

=Φ𝛼z1(n)

Where z1 is defined as

z1

=Φ𝛽0(n)

4.2.3. Φ𝛽2The second Φ𝛽 (Φ𝛽2) is defined

Φ𝛽2(n)

=Φ𝛼z2(n)

Where z2 is defined as

z2

=Φ𝛼z1(n)

4.2.4. Φ𝛽3The third Φ𝛽 (Φ𝛽3) is defined

Φ𝛽3(n)

=Φ𝛼z3(n)

Where z2 is defined as

z3

=Φ𝛼z2(n)

4.2.5. Φ𝛽wThe wth Φ𝛽 (Φ𝛽w) is defined

Φ𝛽w(n)

=Φ𝛼zw(n)

Where zw is defined as

zw

=Φ𝛼zw-1(n)

4.3. Φ𝛾(n)The wth Φ𝛾 (Φ𝛾w) is defined

Φ𝛾w(n)

=Φ𝛽zw-1(n)

Where zw is defined as

zw

=Φ𝛽zw-1(n)

4.4. Greek alphabetWe use the order of Greek alphabet: α,β, γ, δ, ε, ζ, η, θ, ι, κ, λ, μ, ν, ξ, ο, π, ρ, ς, τ, υ, φ, χ, ψ, ω. 4.5. Φ𝜓(n)The wth Φ𝜓 (Φ𝜓w) is defined

Φ𝜓w(n)

=Φ𝜒zw(n)

Where zw is defined as

zw

=Φχzw-1(n)

4.6. Φ𝜔(n)This function is like the next level of Φ function, but we dont have another vatiation phi in greek letter, so I will use the lasti greek leter, 𝜔, to denote the strongest of all, greatest of all, Φ function. There is no wth Φ𝜔. There is only one Φ𝜔(n) that are defined as following

	Φ𝜔(n)	=Φ𝜓(a1)(n)
	a1	=Φ𝜓(a2)(n)
	a2	=Φ𝜓(a3)(n)
	⋮	⋮
	aΦ𝜓n(n)	=Φ𝜓n(n)

This is the pan-ultimate 𝜑 function: a function that so big, I can't really understand it. [Under Construction]