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Re: R: Re: Goodstein expansion

  • To: mathgroup at smc.vnet.net
  • Subject: [mg132338] Re: R: Re: Goodstein expansion
  • From: Matthias Bode <lvsaba at hotmail.com>
  • Date: Wed, 12 Feb 2014 05:13:50 -0500 (EST)
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Hola,

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

In[1]:= 2^6 + 2^4 + 2^2 + 1
2^(2^2 + 2) + 2^2^2 + 2^2 + 2^0

Out[1]= 85
Out[2]= 85

In[3]:= 85 == 87

Out[3]= False

Best regards,

MATTHIAS BODE.
S 17.36398
W 66.218162'590 m. AMSL.



> From: Roberto.Brambilla at rse-web.it
> Subject: R: Re: Goodstein expansion
> To: mathgroup at smc.vnet.net
> Date: Tue, 11 Feb 2014 02:56:42 -0500
>
> 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
>
>
>
>
>
>
> RSE SpA ha adottato il Modello Organizzativo ai sensi del D.Lgs.231/2001, in=
>  forza del quale l'assunzione di obbligazioni da parte della Societ=E0 =
avvie=
> ne con firma di un procuratore, munito di idonei poteri.
> RSE adopts a Compliance Programme under the Italian Law (D.Lgs.231/2001).=
 Ac=
> cording to this RSE Compliance Programme, any commitment of RSE is take=
n by=
>  the signature of one Representative granted by a proper Power of Attorne=
y.=
>  Le informazioni contenute in questo messaggio di posta elettronica sono =
ris=
> ervate e confidenziali e ne e' vietata la diffusione in qualsiasi modo o =
for=
> ma. Qualora Lei non fosse la persona destinataria del presente messaggio=
, La=
>  invitiamo a non diffonderlo e ad eliminarlo, dandone gentilmente comun=
icazi=
> one al mittente. The information included in this e-mail and any attachme=
nts=
>  are confidential and may also be privileged. If you are not the correct =
rec=
> ipient, you are kindly requested to notify the sender immediately, to=
 cancel=
>  it and not to disclose the contents to any other person.
>
 		 	   		  =

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<body class='hmmessage'><div dir='ltr'>Hola,<br><br>"87 = 2^6+2^4+2=
^2+1 = 2^(2^2+2)+2^(2^2)+2^2+2^0":<br><br>In[1]:= 2^6 + 2^4 + 2^2 + 1<b=
r>2^(2^2 + 2) + 2^2^2 + 2^2 + 2^0<br><br>Out[1]= 85<br>Out[2]= 85<br><b=
r>In[3]:= 85 == 87<br><br>Out[3]= False<br><br>Best regards,<br><=
br><strong><em></em></strong><style>=0A=
</style><div><font style="font-size: 10pt;" face="Arial" size="2">M=
ATTHIAS BODE.</font></div><br><div><font style="font-size: 10pt;" face=
="Arial" size="2">S 17.36398<br>W 66.21816</font></div><div><font face=
="Arial" size="2">2'590 m. AMSL.</font></div>=0A=
<div>&nbsp;</div>=0A=
<div><font face="Arial" size="2"><br><a href="mailto:LVSABA at HOTMAIL.C=
OM"></a></font></div><br><br><div>&gt; From: Roberto.Brambilla at rse-web.it=
<br>&gt; Subject: R: Re: Goodstein expansion<br>&gt; To: mat=
hgroup at smc.vnet.net<br>&gt; Date: Tue, 11 Feb 2014 02:56:42 -0500<br>&g=
t; <br>&gt; Many thanks to  Bob Hanlon, Bill Rowe and Murray Eisenber=
g who pay attention to my question.<br>&gt; <br>&gt; I have to add some=
 points.<br>&gt; 1)  the expansion I called Goodstein must be called more=
 correctly Cantor expansion.<br>&gt; 2) Cantor expansion is the usual bas=
e b power expansion where also exponents (if the case when &gt;b)<br>&gt=
; must be similarly expanded as in the example <br>&gt; <br>&gt; 87 =
= 2^6+2^4+2^2+1 = 2^(2^2+2)+2^(2^2)+2^2+2^0<br>&gt; <br>&gt; where =
6 and 4 are expanded too ("graphically" you must see only b's<br>&gt; or =
numbers less than b).<br>&gt; The Goodstein sequence is essentially obtai=
ned increasing base b to b+1  :<br>&gt; <br>&gt; GoodsteinProcess : Giv=
en a number m1 , write m1 in base-2-CantorForm.<br>&gt; Replace 2 with =
3 and obtain a new number n1. <br>&gt; Expand m2=n1-1  in base-3-Cantor=
Form<br>&gt; Replace 3 with 4 and obtain a new number n2<br>&gt; Expand=
 m3=n2-1  in base-4-CantorForm etc..<br>&gt; <br>&gt; Property: this =
process terminates at 0 for any starting m1.<br>&gt; <br>&gt; I am aski=
ng an help in building a  CantorForm[m,b]  procedure in a simple way,<b=
r>&gt; when exponents becomes large and they also have to be expanded. <b=
r>&gt; <br>&gt; Regards, Roberto<br>&gt; <br>&gt; <br>&gt; <br>=
&gt; <br>&gt; <br>&gt; <br>&gt; RSE SpA ha adottato il Modello Orga=
nizzativo ai sensi del D.Lgs.231/2001, in=<br>&gt;  forza del quale l=
'assunzione di obbligazioni da parte della Societ=E0 avvie=<br>&gt; n=
e con firma di un procuratore, munito di idonei poteri.<br>&gt; RSE ado=
pts a Compliance Programme under the Italian Law (D.Lgs.231/2001). Ac=<br=
>&gt; cording to this RSE Compliance Programme, any commitment of RSE i=
s taken by=<br>&gt;  the signature of one Representative granted by a p=
roper Power of Attorney.=<br>&gt;  Le informazioni contenute in questo =
messaggio di posta elettronica sono ris=<br>&gt; ervate e confidenziali=
 e ne e' vietata la diffusione in qualsiasi modo o for=<br>&gt; ma. Qua=
lora Lei non fosse la persona destinataria del presente messaggio, La=<=
br>&gt;  invitiamo a non diffonderlo e ad eliminarlo, dandone gentilmen=
te comunicazi=<br>&gt; one al mittente. The information included in thi=
s e-mail and any attachments=<br>&gt;  are confidential and may also be=
 privileged. If you are not the correct rec=<br>&gt; ipient, you are =
kindly requested to notify the sender immediately, to cancel=<br>&gt;=
  it and not to disclose the contents to any other person.<br>&gt; <br></=
div> 		 	   		  </div></body>
</html>=

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