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

((10^{10^100})^^8)^{((10^3003)^(10^3003)+10^100)}+(10^{10^10^10^10^100})^{10^6000^10^273}

> ((10^{10^100})^^8)^{((10^3003)^(10^3003)+10^100)}

> ((10^{10^100})^^8)

= (10^{10^100})^(10^{10^100})^(10^{10^100})^(10^{10^100})^(10^{10^100})^(10^{10^100})^(10^{10^100})^(10^{10^100})

> 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

((10^{10^100})^^8)^{((10^3003)^(10^3003)+10^100)}+(10^{10^10^10^10^100})^{10^6000^10^273}

We'll need to handle this in parts. First let's look at just ((10^{10^100})^^8).

(10^{10^100})^^1 = 10^{10^100} = E100#2

(10^{10^100})^^2 = (10^{10^100})^{10^10^100}

= 10^{10^100*10^10^100}

= 10^{10^(100+10^100)}

= E(100+E100)#2

(10^{10^100})^^3 = (10^{10^100})^(10^{10^100})^(10^{10^100})

= (10^{10^100})^{10^10^(100+10^100)}

= 10^{10^100*10^10^(100+10^100)}

= 10^{10^(100+10^(100+10^100))}

< 10^{10^(2*10^(100+10^100))}

< 10^{10^(10^(101+10^100)))}

= E(101+E100)#3

(10^{10^100})^^4 = (10^{10^100})^(10^{10^100})^(10^{10^100})^(10^{10^100})

< (10^{10^100})^{10^10^10^(101+10^100)}

^{= 1010^100*10^10^10^(101+10^100)}

= 10^{10^(100+10^10^(101+10^100))}

< 10^{10^(10*10^10^(101+10^100))}

= 10^{10^(10^(1+10^(101+10^100)))}

< 10^{10^(10^(10*10^(101+10^100)))}

= 10^{10^(10^(10^(1+101+10^100)))}

= 10^{10^(10^(10^(102+10^100)))}

= E(102+E100)#4

similarly:

(10^{10^100})^^5 < E(104+E100)#5

(10^{10^100})^^6 < E(105+E100)#6

(10^{10^100})^^7 < E(106+E100)#7

(10^{10^100})^^8 < E(107+E100)#8

Now let's deal with ((10^{10^100})^^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:

((10^{10^100})^^8)^{((10^3003)^(10^3003)+10^100)}

< (E(107+E100)#8)^{10^10^3008}

= 10^{(E(107+E100)#7)*10^10^3008)}

= 10^{10^(E(107+E100)#6+10^3008))}

< 10^{10^(2*E(107+E100)#6)}

= 10^{10^(2*10^10^10^10^10^10^(107+10^100))}

< 10^{10^(10^(1+10^10^10^10^10^(107+10^100)))}

....

<< 10^{10^10^10^10^10^10^10^(108+10^100)}

= E(108+E100)#8

And now for (10^{10^10^10^10^100})^{10^6000^10^273}:

(10^{10^10^10^10^100})^{10^6000^10^273}

= 10^{(10^10^10^10^100*10^6000^10^273)}

= 10^{10^(10^10^10^100+6000^10^273))}

< 10^{10^(2*10^10^10^100)}

<< 10^{10^10^10^10^101}

= E101#5

And now for the whole sum:

((10^{10^100})^^8)^{((10^3003)^(10^3003)+10^100)}+(10^{10^10^10^10^100})^{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.