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The giggol is a 1 followed by a 1 followed by a 1 etc etc and repeating this proces 99 times then the sentence ends by .... followed by a 1 with 10^10 zero's zero's zero's......

So therefore i would write it as 1>(1>(1>...99times the 1> term.....(1>10)..99braces..))


It's more easy to show it with the decker number that is in arrow notation (10,10,2)

is exactly

10^10^10^10^10^10^10^10^10^10=

10^......8 tens before.......^(1>10)=

10^......7 tens before.......^(1>(1>10))=

10^.....1 ten before........^(1>(1>(1>(1>(1>(1>(1>(1>10))))))))=

9 times the term "(1>" and then >10 and 9 braces.


So it also works out for the giggol in arrow notation defined as (10,100,2)

99 times the term "(1>" and then >10 and 99 braces


Now

E decker  therefore is

E 1>1>1>1>1>1>1>1>1>10 =

1>(1>1>1>1>1>1>1>1>10) > (1>1>1>1>1>1>1>1>10)


To calculate this one we first look at the patterns of smaller similar E powers like

E 1>1>10 or E 1>1>1>10


E 1>10 = 1>1>11

E 1>1>10 = 1>1>((1>8)~10)

E 1>1>1>10 = 1>1000.....1>10.....>1000.....1>10.....=

                        1>1>                        (1000.....1>10.... + 1>10)=

                        1>1>                        ((1> ((1>10)-(10+1)))~(1>10))=

                        1>1>                        ((1> 9 999 999 989)~(1>10))=

                        1>(1>((1>(9re8~89))~(1>10)))


Now i use a new notation to indicate multiple "(>1" term with

(m)^>1

so for example 1>1>1>10 then is (3^>1)>10


so

E (4^>1)>10 = 1>(1000.....1>1>10.....>10000.....1>1>10......) =

                        1>1>1>(1>1>10 - 1>10)~1>1>10=

                        1>1>1>(9re(1>10-(10+1)))~8~(9re10)~(1>(1>10))=

                        1>(1>((1>((9re(9re8~89))~8~(9re10)))~(1>(1>10)))) 


What becomes clear already is that

E 1>(m^>1)>10 = 1>(1>(1>((1>((((m-3)^>1)>10) - (1+(1>(((m-4)^>1)>10))))) ~ (1>(((m-3)^>1)>10))))

as long as m=4 or more

so the only 'difficult' part in the number formation is the part of

(1>(((m-1)^>1) >10)) - (1+ (1>(((m-2)^>1)>10))))


We need to find it for m=9 and then for the giggol for m=99.

Lets try m=9;

As the pattern already shows

it becomes

1>1>1>(9re(1>((m-4)^>1>10)-(1+(1>m-5^>1>10))))~8~(9re1>(m-5)^>1>10)

so actually only 

m=9 has to be placed in the equation here;

9re((1>((9-4)^>1)>10)-(1+((1>(9-5)^>1)>10)=

9re(1>1>1>1>1>10 - (1>1>1>1>10 + 1)) =

9re (9re (1>1>1>1>10 - (1>1>1>10 + 1)))~8~9re1>1>1>10 =

9re (9re 9re (1>1>1>10 -     (1>1>10 + 1)) ~8~9re1>1>10)~8~9re1>1>1>10=

9re 9re 9re 9re 1>1>10 -           1>10 + 1))~8~9re1>10)~8~9re1>1>10.....=

9re (((9re (((9re (((9re ((9re ((9re8)~89))~(8~(9re10))))~(8~(9re(1>10)))))~(8~(9re(1>(1>10))))))~(8~(9re(1>(1>(1>10))))))


Thus on a list from E 1>1 to E 1>((m)^>10) ;

E 1>1 = 1>(1>1)

E 1>(1>1) =

E 1>(1^>1)>1 = 1>(1>11)

E 1>(1>(1>1) =

E 1>(2^>1)>1 = 1>(1>((1>8)~(1>1)))

E 1>(3^>1)>1 = 1>(1>((1>((9re8)~8~(9re1)))~(1>(1^>1)>1)))

E 1>(4^>1)>1 = 1>(1>(((1>((9re                    ((9re8)~8~(9re1)))        ~8~(9re            (1>1))))~(1>(2^>1)>1)))

E 1>(5^>1)>1 = 1>(1>(((1>((9re(9re             ((9re8)~8~(9re1))))       ~8~(9re(1>(1^>1)>1))))~(1>(3^>1)>1)))

E 1>(6^>1)>1 = 1>(1>(((1>((9re(9re(9re       ((9re8)~8~(9re1)))))      ~8~(9re(1>(2^>1)>1))))~(1>(4^>1)>1)))

now i create for the ......9re(9re(9re(9..... part the notation ['(9re're(m)]~~  so for E 1>(6^>1)>1 i now write

            1>(1>(((1>(['(9re're3]~~       ((9re8)~8~(9re1)))))..............

also with braces i can do that in the form of ['('re(m)] what can be handy if the number of braces becomes really big like over the 30.

in the case of E 1>(6^>1)>1 i write it as

           1>(1>(((1>(['(9re're3]~~       ((9re8)~8~(9re1~~[')'re5] ~8~(.............


So the formula of E 1>((m)^>1)>1 can be formulated to be

1>(1>(((1>(['(9re're(m-3)]~~((9re8)~8~(9re1~~[')'re(m-1)]~8~(9re(1>((m-4)^>1)>1))))~(1>((m-2)^>1)>1)))

for a giggol is equal to

1>(100^>1)>1 m is equal to 100 so

E giggol = 

E (10,100,2) = 1>(1>(((1>['(9re're97]~~((9re8)~8~(9re1~~[')'re99]~8~(9re(1>(96^>1)>1))))~(1>(98^>1)>1)))

and for the decker;

E decker = 

E (10,10,2) = 1>(1>(((1>(['9re're  7]~~((9re8)~8~(9re1~~ [')'re  9]~8~(9re(1>(  6^>1)>1))))~(1>((  8^>1)>1)))

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