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R: Re: Goodstein expansion

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  • Subject: [mg132329] R: Re: Goodstein expansion
  • From: "Brambilla Roberto Luigi (RSE)" <Roberto.Brambilla at>
  • Date: Tue, 11 Feb 2014 02:56:42 -0500 (EST)
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Many thanks to  Bob Hanlon, Bill Rowe and Murray Eisenberg who pay attention to my question.

I have to add some points.
1)  the expansion I called Goodstein must be called more correctly Cantor expansion.
2) Cantor expansion is the usual base b power expansion where also exponents (if the case when >b)
must be similarly expanded as in the example 

87 = 2^6+2^4+2^2+1 = 2^(2^2+2)+2^(2^2)+2^2+2^0

where 6 and 4 are expanded too ("graphically" you must see only b's
or numbers less than b).
The Goodstein sequence is essentially obtained increasing base b to b+1  :

GoodsteinProcess : Given a number m1 , write m1 in base-2-CantorForm.
Replace 2 with 3 and obtain a new number n1. 
Expand m2=n1-1  in base-3-CantorForm
Replace 3 with 4 and obtain a new number n2
Expand m3=n2-1  in base-4-CantorForm etc..

Property: this process terminates at 0 for any starting m1.

I am asking an help in building a  CantorForm[m,b]  procedure in a simple way,
when exponents becomes large and they also have to be expanded. 

Regards, Roberto

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