genu's numbers proof
This is the subpage of my number list where I prove lower and upper bounds for the two Genu's numbers.
Genu's Number I
Lower Bound
((1010^100)^^8)((10^3003)^(10^3003)+10^100)+(1010^10^10^10^100)10^6000^10^273
> ((1010^100)^^8)((10^3003)^(10^3003)+10^100)
> ((1010^100)^^8)
= (1010^100)^(1010^100)^(1010^100)^(1010^100)^(1010^100)^(1010^100)^(1010^100)^(1010^100)
> 10^10^10^10^10^10^10^10^10^100
= E100#9
This means that Genu's Number > E100#9 (aka googoloctiplex). However, this isn't as much of an underestimate as you may think, which you will see as I upper-bound the number.
Upper Bound
((1010^100)^^8)((10^3003)^(10^3003)+10^100)+(1010^10^10^10^100)10^6000^10^273
We'll need to handle this in parts. First let's look at just ((1010^100)^^8).
(1010^100)^^1 = 1010^100 = E100#2
(1010^100)^^2 = (1010^100)10^10^100
= 1010^100*10^10^100
= 1010^(100+10^100)
= E(100+E100)#2
(1010^100)^^3 = (1010^100)^(1010^100)^(1010^100)
= (1010^100)10^10^(100+10^100)
= 1010^100*10^10^(100+10^100)
= 1010^(100+10^(100+10^100))
< 1010^(2*10^(100+10^100))
< 1010^(10^(101+10^100)))
= E(101+E100)#3
(1010^100)^^4 = (1010^100)^(1010^100)^(1010^100)^(1010^100)
< (1010^100)10^10^10^(101+10^100)
= 1010^100*10^10^10^(101+10^100)
= 1010^(100+10^10^(101+10^100))
< 1010^(10*10^10^(101+10^100))
= 1010^(10^(1+10^(101+10^100)))
< 1010^(10^(10*10^(101+10^100)))
= 1010^(10^(10^(1+101+10^100)))
= 1010^(10^(10^(102+10^100)))
= E(102+E100)#4
similarly:
(1010^100)^^5 < E(104+E100)#5
(1010^100)^^6 < E(105+E100)#6
(1010^100)^^7 < E(106+E100)#7
(1010^100)^^8 < E(107+E100)#8
Now let's deal with ((1010^100)^^8)((10^3003)^(10^3003)+10^100), first just the exponent
(10^3003)(10^3003)+10^100:
(10^3003)(10^3003)+10^100
= 10(3003*10^3003)+10^100
< 10(10,000*10^3003)+10^100
= 10(10^4*10^3003)+10^100
= 10(10^3007)+10^100
< 2*10(10^3007)
< 10(10^3008)
So:
((1010^100)^^8)((10^3003)^(10^3003)+10^100)
< (E(107+E100)#8)10^10^3008
= 10(E(107+E100)#7)*10^10^3008)
= 1010^(E(107+E100)#6+10^3008))
< 1010^(2*E(107+E100)#6)
= 1010^(2*10^10^10^10^10^10^(107+10^100))
< 1010^(10^(1+10^10^10^10^10^(107+10^100)))
....
<< 1010^10^10^10^10^10^10^(108+10^100)
= E(108+E100)#8
And now for (1010^10^10^10^100)10^6000^10^273:
(1010^10^10^10^100)10^6000^10^273
= 10(10^10^10^10^100*10^6000^10^273)
= 1010^(10^10^10^100+6000^10^273))
< 1010^(2*10^10^10^100)
<< 1010^10^10^10^101
= E101#5
And now for the whole sum:
((1010^100)^^8)((10^3003)^(10^3003)+10^100)+(1010^10^10^10^100)10^6000^10^273
< E(108+E100)#8+E101#5
< 2*E(108+E100)#8
<< E(109+E100)#8
<<< E(E101)#8
= E101#9
Therefore:
E100#9 << Genu's Number << E101#9
Genu's Number II
Upper Bound
70!2*35!2*812,500*812,500812,500
= 70^^69^^68^^67.......^^4^^3^^2^^1 * 35!2*812,500*812,500812,500
= 70^^69^^68^^67.......^^4^^3^^2 * 35!2*812,500*812,500812,500
= 70^^69^^68^^67.......^^4^^3^3 * 35!2*812,500*812,500812,500
= 70^^69^^68^^67.......^^4^^27 * 35!2*812,500*812,500812,500
< 100^^100^^100^^100........^^100^^100 (68 100s) * 35!2*812,500*812,500812,500
= 100^^^68 * 35!2*812,500*812,500812,500
< (10^^^2)^^^68 * 35!2*812,500*812,500812,500
< 10^^^70 * 35!2*812,500*812,500812,500
< 10^^^70 * 35!2*812,500*812,500812,500
= 10^^^70 * 35^^34^^33^^32.......^^4^^3^^2^^1 * 812,500*812,500812,500
= 10^^^70 * 35^^34^^33^^32.......^^4^^27 * 812,500*812,500812,500
< 10^^^70 * 100^^100^^100^^100.......^^100^^100 (33 100s) * 812,500*812,500812,500
= 10^^^70 * 100^^^33 * 812,500*812,500812,500
< 10^^^70 * (10^^^2)^^^33 * 812,500*812,500812,500
< 10^^^70 * 10^^^35 * 812,500*812,500812,500
= 10^^^70 * 10^^^35 * 812,500812,501
< 10^^^70 * 10^^^35 * 1,000,0001,000,000
= 10^^^70 * 10^^^35 * 106,000,000
= 10^^^70 * 10^^(10^^^35) * 106,000,000
= 10^^^70 * 10^(10^^(10^^^35-1)) * 106,000,000
= 10^^^70 * 10^(10^^(10^^^35-1)+6,000,000)
< 10^^^70 * 10^(2*10^^(10^^^35-1))
< 10^^^70 * 10^(10^(10^^(10^^^35-1)))
= 10^^^70 * 10^(10^^(10^^^35))
= 10^^(10^^^69) * 10^(10^^(10^^^35))
= 10^(10^^(10^^^69-1)) * 10^(10^^(10^^^35))
= 10^(10^^(10^^^69-1)+(10^^(10^^^35)))
< 10^(2*10^^(10^^^69-1))
< 10^(10^(10^^(10^^^69-1)))
= 10^(10^^(10^^^69))
= 10^^(10^^^69+1)
<< 10^^(10^^^70)
= 10^^^71
So the whole number is MUCH LESS than 10^^^71.
Upper Bound
Genu's Number II = 70^^69^^68^^67.......^^4^^3^^2^^1 * 35!2*812,500*812,500812,500
> 70^^69^^68^^67.......^^4^^3^^2^^1 (we can ignore the multiplication on the right)
> 10^^10^^10^^10^^......^^10^^9^^8^^7^^6^^5^^4^^3^^2^^1 (61 10's)
= 10^^10^^10^^10^^......^^10^^9^^8^^7^^6^^5^^4^^27
> 10^^10^^10^^10^^......^^10^^9^^8^^7^^6^^5^^(4^^2)^^25
> 10^^10^^10^^10^^......^^10^^9^^8^^7^^6^^5^^10^^25
> 10^^10^^10^^10^^......^^10^^9^^8^^7^^6^^(5^^2)^^(10^^25-2)
> 10^^10^^10^^10^^......^^10^^9^^8^^7^^6^^10^^(10^^25-2)
> 10^^10^^10^^10^^......^^10^^9^^8^^7^^6^^10^^(10^^24)
> 10^^10^^10^^10^^......^^10^^9^^8^^7^^(6^^2)^^(10^^(10^^24-2))
> 10^^10^^10^^10^^......^^10^^9^^8^^7^^10^^10^^10^^23
with a similar technique:
> 10^^10^^10^^10^^......^^10^^9^^8^^10^^10^^10^^10^^22
> 10^^10^^10^^10^^......^^10^^9^^10^^10^^10^^10^^10^^21
> 10^^10^^10^^10^^......^^10^^10^^10^^10^^10^^10^^10^^20
> 10^^10^^10^^10^^......^^10^^10^^10^^10^^10^^10^^10^^10
= 10^^^68
Therefore:
10^^^68 << Genu's Number II << 10^^^71. This also shows that Genu's number is much less than a gaggol.