Fast-growing hiearchy: How fast grows f_(ω+2) ?

《§》carbonara <3〖ƧƐ〗
Owner

Maths Lover <3 <3 <3

10000000000000000000 times stronger than pure sulfuric acid ( ͡° ͜ʖ ͡°)

Posts: 912
Favorite Acid: Trifluoromethanesulfonic Acid
Favorite Scientist: P1kachu
Favorite Admin: Miguel888

Fast-growing hiearchy: How fast grows f_(ω+2) ? Jun 14, 2015 10:31:07 GMT Kruso/iShark likes this

Quote

Post by 《§》carbonara <3〖ƧƐ〗 on Jun 14, 2015 10:31:07 GMT

Soo, let's start:

f₀(n) =n+1
f_a+1(n) = f_aⁿ(n)
f_a(n) = f_a(n)(n) iff a is a limit ordinal.

Therefore:

f₁(n) = 2n
f₂(n) > 2ⁿ = 2^2
f₃(n) > ⁿ2 = 2^^2 (2 tetrated to n) = 2^{2^{2^{2^{2^...²}}}} n times
f₄(n) > 2^^^n = ^{^{^{^{^{^²...2}2}2}2}2}2 (n times) = 2^{2^{2^...²}} (2^{2^{2^...²}} (2^{2^{2^...²}} (etc...) times) times (with n layers - crazily big!!!)
f_a(n) > 2^^^...(n-1) arrows...^^^n for a < ω

ω = lim_n->+infinity(n)

According to rule 3, f_ω(n)=f_ω(n)(n)
(We take the n-th member of the ω's fundamental sequence)
<=>f_ω(n)=f_n(n)

Theorem: f_ω(n) grows faster than any primitive-recursive function (any g(x)=a^^^^...^^^^b with a fixed number of arrows)

According to rule 2, f_ω+1[/sup](n)=fⁿ_ω(n)
F.e., f_=ω+1(5)=f_ω(f_ω(f_ω(f_ω(f_ω(5)))))=f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))

f_ω+2(5) = f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))))}(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))))}(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))
ABANDON ALL HOPE!!!! xD

f_ω+3(n)'s growth rate is beyond imagination!!!! And incredibly slow compared to f_ω+ω(n)...

Last Edit: Jun 17, 2015 10:31:20 GMT by 《§》carbonara <3〖ƧƐ〗

P1kachu
Administrator

Tetramethyldiaminobenzhydrylphosphinous Acid

P1kachu used Flamethrower! It's super effective!

Posts: 1,076

Favorite Admin: Carbonic Acid Cookie
Favorite Moderator: Hi Y3ti
Most Hated Admin: P1kachu
Mini-Profile Text Color: ffffff
Mini-Profile Name Color: ffffff
Mini-Profile Background: {"image":"http://www.inspirationluv.com/wp-content/uploads/2015/03/iphone-wallpaper-textures.jpg","color":""}

Fast-growing hiearchy: How fast grows f_(ω+2) ? Jun 14, 2015 11:06:05 GMT

Quote

Post by P1kachu on Jun 14, 2015 11:06:05 GMT

Jun 14, 2015 10:31:07 GMT 《§》carbonara <3〖ƧƐ〗 said:

f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))))}(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))))}(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))

How did you do type this all

My name should have been Tetramethyldiaminobenzhydrylphosphinous Acid, but it only accepts 30 and below letters

《§》carbonara <3〖ƧƐ〗
Owner

Maths Lover <3 <3 <3

10000000000000000000 times stronger than pure sulfuric acid ( ͡° ͜ʖ ͡°)

Posts: 912
Favorite Acid: Trifluoromethanesulfonic Acid
Favorite Scientist: P1kachu
Favorite Admin: Miguel888

Fast-growing hiearchy: How fast grows f_(ω+2) ? Jun 14, 2015 11:30:08 GMT

Quote

Post by 《§》carbonara <3〖ƧƐ〗 on Jun 14, 2015 11:30:08 GMT

Jun 14, 2015 11:06:05 GMT P1kachu said:

Jun 14, 2015 10:31:07 GMT 《§》carbonara <3〖ƧƐ〗 said:

f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))))}(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))))}(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))

How did you do type this all

copy&paste m8

P1kachu
Administrator

Tetramethyldiaminobenzhydrylphosphinous Acid

P1kachu used Flamethrower! It's super effective!

Posts: 1,076

Favorite Admin: Carbonic Acid Cookie
Favorite Moderator: Hi Y3ti
Most Hated Admin: P1kachu
Mini-Profile Text Color: ffffff
Mini-Profile Name Color: ffffff
Mini-Profile Background: {"image":"http://www.inspirationluv.com/wp-content/uploads/2015/03/iphone-wallpaper-textures.jpg","color":""}

Fast-growing hiearchy: How fast grows f_(ω+2) ? Jun 14, 2015 11:30:39 GMT

Quote

Post by P1kachu on Jun 14, 2015 11:30:39 GMT

Jun 14, 2015 11:30:08 GMT 《§》carbonara <3〖ƧƐ〗 said:

Jun 14, 2015 11:06:05 GMT P1kachu said:

How did you do type this all

copy&paste m8

My name should have been Tetramethyldiaminobenzhydrylphosphinous Acid, but it only accepts 30 and below letters

《§》carbonara <3〖ƧƐ〗
Owner

Maths Lover <3 <3 <3

10000000000000000000 times stronger than pure sulfuric acid ( ͡° ͜ʖ ͡°)

Posts: 912
Favorite Acid: Trifluoromethanesulfonic Acid
Favorite Scientist: P1kachu
Favorite Admin: Miguel888

Fast-growing hiearchy: How fast grows f_(ω+2) ? Jun 14, 2015 11:31:48 GMT

Quote

Post by 《§》carbonara <3〖ƧƐ〗 on Jun 14, 2015 11:31:48 GMT

Jun 14, 2015 11:30:39 GMT P1kachu said:

Jun 14, 2015 11:30:08 GMT 《§》carbonara <3〖ƧƐ〗 said:

copy&paste m8

its the actual value of f_w+2(5). Big enough? ( ͡° ͜ʖ ͡°)

P1kachu
Administrator

Tetramethyldiaminobenzhydrylphosphinous Acid

P1kachu used Flamethrower! It's super effective!

Posts: 1,076

Favorite Admin: Carbonic Acid Cookie
Favorite Moderator: Hi Y3ti
Most Hated Admin: P1kachu
Mini-Profile Text Color: ffffff
Mini-Profile Name Color: ffffff
Mini-Profile Background: {"image":"http://www.inspirationluv.com/wp-content/uploads/2015/03/iphone-wallpaper-textures.jpg","color":""}

Fast-growing hiearchy: How fast grows f_(ω+2) ? Jun 14, 2015 11:34:23 GMT

Quote

Post by P1kachu on Jun 14, 2015 11:34:23 GMT

The answer is:

f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))))}(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))))}(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}⁵(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))))

Last Edit: Jun 14, 2015 11:35:08 GMT by P1kachu

My name should have been Tetramethyldiaminobenzhydrylphosphinous Acid, but it only accepts 30 and below letters

Kruso/iShark
Arithmetical Transfinite Recursion

Cataclysmic GD Player ( ͡° ͜ʖ ͡°)

Cataclysmic GD Player!

Posts: 118

Favorite Admin: P1kachu
Favorite Moderator: Hi Y3ti
Mini-Profile Text Color: db1c2b
Mini-Profile Name Color: db1c2b

Fast-growing hiearchy: How fast grows f_(ω+2) ? Jun 14, 2015 11:37:42 GMT

Quote

Post by Kruso/iShark on Jun 14, 2015 11:37:42 GMT

Jun 14, 2015 11:34:23 GMT P1kachu said:

The answer is:

f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_{f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f_{f_{f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))))}(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)}(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))))}(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))))}(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))}⁵(f_{f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5)))}(f_{f_f₅(5)(f₅(5))}(f_f₅(5)(f₅(5))))))

Congratz!

Time is evwything 2 a shark!