# 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.