Googology Wiki
Googology Wiki

X-Sequence Hyper-Exponential Notation is an extension of the hyper operators by SuperJedi224[1], partly inspired by both Cascading-E notation and BEAF.


Main Definition

m{1}n = mn

m{α}1 = m

m{α+1}(n+1) = m{α}(m{α+1}n)

If α is a limit case: m{α}n = m{α[n]}m

Definition of α[n]

X[n] = n

(α+β)[n] = α+β[n]

(α*(β+1))[n] = α*β+α[n]

(α*β)[n] = α*β[n], if β is a limit case

α*1 = α

β+1)[n] = αβ*α[n]

β)[n] = αβ[n], if β is a limit case

α1 = α

(α↑↑X)[0] = 1

(α↑↑X)[n] = α(α↑↑X)[n-1]

α>(β+1) = (α>β)↑↑X

(α>β)[n] = α>β[n], if β is a limit case.

α>0 = α



Growth Rate

This notation is believed to have a limit ordinal of \(\zeta_0\) in the Fast-growing hierarchy.


4{3}7 = 4{2}4{2}4{2}4{2}4{2}4{2}4 (this is solved from right to left).

4{X+1}3 = 4{X}4{X}4 = 4{X}4{4}4 = 4{4{4}4}4.

4{X>X>X}3 = 4{X>X>3}4 = 4{X>((X↑↑X)↑↑X)↑↑X)}4

In Other Notations

a{c}b = \(a\uparrow^cb\)

a{X}b = \(a\uparrow^ba\)

a{X+1}b = {a,b,1,2}

a{X+2}b = {a,b,2,2}

a{X*2}b= {a,a,b,2}