Tethrathor

Tethrathor is equal to E100#^(#^^#*#)100 in Extended Cascading-E Notation. The term was coined by Sbiis Saibian. This number belongs to the tethrathoth regiment.
== Etymology ==
The name of this number is based on the number "tethrathoth" and the number "godgathor".
== Difference between hecato-tethrathoth ==
Tethrathor and hecato-tethrathoth (tethrafact) represent very similar values, but with slightly different values, as (#^^#)^# and #^(#^^#*#) represent the same ordinals in the fast-growing hierarchy, and the ordinal \(\omega^{\omega^{\varepsilon_0+1}} = \varepsilon_0^{\omega}\). In order to compute the difference between tethrathor and hecato-tethrathoth, it can be computed as follows:
# #^(#^^X) delimiter is equal to #^^(X+1). The +1 is an additional offset. Each #^(#^^#) will increase the height of the hyperion power tower of #^^# from the bottom by one #.
# Let #-1 to be defined such that #-1[n] = #[n]-1, treating # as a indefinite value.
# Hecato-tethrathoth is equal to E100#^(#^^(#-1)*#)100. It is easy to show that #-1 is slightly smaller than just #.
# Hecato-tethrathoth can be solved by:
#* E100#^(#^^(#-1)*#)100
#* = E100(#^(#^^(#-1)))^(100)100
#* = E100(#^(#^^(#-1))^(99)*#^(#^^(#-1))100
#* = E100(#^(#^^(#-1))^(99)*#^(#^^(100-1))100
#* = E100(#^(#^^(#-1))^(99)*#^(#^^(99))100
#* = E100(#^(#^^(#-1))^(99)*#^^(100)100
# Similarly, tethrathor can also be solved using the similar method as hecato-godgathor (godgathorfact) by:
#* E100#^(#^^#*#)100
#* = E100#^(#^^#)*#^(#^^#)*...*#^(#^^#)*#^(#^^#)100 (with 100 #^(#^^#)'s)
#* = E100#^(#^^#)*#^(#^^#)*...*#^(#^^#)*#^(#^^100)100
#* = E100#^(#^^#)*#^(#^^#)*...*#^(#^^#)*(#^^101)100
#* While hecato-tethrathoth can also be expanded in simpler expression, E100#^^#*#^^#*...*#^^#*#^^#100 (with 100 #^^#'s) = E100#^^#*#^^#*...*#^^#*(#^^100)100. We can see the difference that the final hyper-product of the delimiter is #^^100 for hecato-tethrathoth, and #^^101 for tethrathor.
Therefore, tethrathor is slightly larger than hecato-tethrathoth.
== Approximations in other notations ==
Notation 	Approximation
BEAF 	\(\{100,100(X \uparrow\uparrow X*X) 2\}\)Using particular notation \(\{a,b (A) 2\} = A\ \&\ a\) with prime b.
Bird's array notation 	\(\{100,101 [2 \backslash 2] 2\}\)
Hyperfactorial array notation	\(100![1,1,[1],1,2]\)
Fast-growing hierarchy 	\(f_{\varepsilon_0^\omega}(100)\)
Hardy hierarchy 	\(H_{\varepsilon_0^{\varepsilon_0^\omega}}(99)\)
Slow-growing hierarchy 	\(g_{\vartheta(\varepsilon_{\Omega2}^\omega)}(100)\)
== Sources ==
== See also ==
Category:Numbers
Category:Numbers by Sbiis Saibian
Category:Extended Cascading-E numbers
Category:Epsilon level
Category:-thor
Category:Tethrafact series
Category:Tethrathoth — Monster-Giant