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. > = --_6b097b34-4885-4725-883b-c61fdede4fc2_ Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable <html> <head> <style><!-- .hmmessage P { margin:0px; padding:0px } body.hmmessage { font-size: 12pt; font-family:Calibri } --></style></head> <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> </div>=0A= <div><font face="Arial" size="2"><br><a href="mailto:LVSABA at HOTMAIL.C= OM"></a></font></div><br><br><div>> From: Roberto.Brambilla at rse-web.it= <br>> Subject: R: Re: Goodstein expansion<br>> To: mat= hgroup at smc.vnet.net<br>> Date: Tue, 11 Feb 2014 02:56:42 -0500<br>&g= t; <br>> Many thanks to Bob Hanlon, Bill Rowe and Murray Eisenber= g who pay attention to my question.<br>> <br>> I have to add some= points.<br>> 1) the expansion I called Goodstein must be called more= correctly Cantor expansion.<br>> 2) Cantor expansion is the usual bas= e b power expansion where also exponents (if the case when >b)<br>>= ; must be similarly expanded as in the example <br>> <br>> 87 = = 2^6+2^4+2^2+1 = 2^(2^2+2)+2^(2^2)+2^2+2^0<br>> <br>> where = 6 and 4 are expanded too ("graphically" you must see only b's<br>> or = numbers less than b).<br>> The Goodstein sequence is essentially obtai= ned increasing base b to b+1 :<br>> <br>> GoodsteinProcess : Giv= en a number m1 , write m1 in base-2-CantorForm.<br>> Replace 2 with = 3 and obtain a new number n1. <br>> Expand m2=n1-1 in base-3-Cantor= Form<br>> Replace 3 with 4 and obtain a new number n2<br>> Expand= m3=n2-1 in base-4-CantorForm etc..<br>> <br>> Property: this = process terminates at 0 for any starting m1.<br>> <br>> I am aski= ng an help in building a CantorForm[m,b] procedure in a simple way,<b= r>> when exponents becomes large and they also have to be expanded. <b= r>> <br>> Regards, Roberto<br>> <br>> <br>> <br>= > <br>> <br>> <br>> RSE SpA ha adottato il Modello Orga= nizzativo ai sensi del D.Lgs.231/2001, in=<br>> forza del quale l= 'assunzione di obbligazioni da parte della Societ=E0 avvie=<br>> n= e con firma di un procuratore, munito di idonei poteri.<br>> RSE ado= pts a Compliance Programme under the Italian Law (D.Lgs.231/2001). Ac=<br= >> cording to this RSE Compliance Programme, any commitment of RSE i= s taken by=<br>> the signature of one Representative granted by a p= roper Power of Attorney.=<br>> Le informazioni contenute in questo = messaggio di posta elettronica sono ris=<br>> ervate e confidenziali= e ne e' vietata la diffusione in qualsiasi modo o for=<br>> ma. Qua= lora Lei non fosse la persona destinataria del presente messaggio, La=<= br>> invitiamo a non diffonderlo e ad eliminarlo, dandone gentilmen= te comunicazi=<br>> one al mittente. The information included in thi= s e-mail and any attachments=<br>> are confidential and may also be= privileged. If you are not the correct rec=<br>> ipient, you are = kindly requested to notify the sender immediately, to cancel=<br>>= it and not to disclose the contents to any other person.<br>> <br></= div> </div></body> </html>= --_6b097b34-4885-4725-883b-c61fdede4fc2_--