branching in p-morphics
David Wilson
davidwwilson at attbi.com
Mon Sep 23 17:26:33 CEST 2002
The short answer is no, the automorphics do not experience the
branching phenomenon observed with the trimorphics.
Sequences of automorphic numbers correspond to 10-adic solutions
to n^2 = n. It can be shown that there are exactly four 10-adic
solutions to this equation: 0, 1, and two other solutions x and y.
The solutions 0 and 1 correspond to the uninteresting automorphic
sequences 0,0,0,0,0,... and 1,0,0,0,0,...
x and y correspond to the interesting solutions 5,2,6,0,9,...
and 6,7,3,9,0,... (that is, sequences A018247 and A018248). These
sequences actually give the 10-adic digits of x and y.
The trimorphic numbers correspond to 10-adic solutions to n^3 = n.
Unlike n^2 = n, this equation has an infinitude of 10-adic
solutions, each yielding a different trimorphic sequence. We
don't want to add an infinite number of sequences to the OEIS.
The infinite number of solutions expresses itself as the branching
phenomenon you described. n^2 = n, with a finite number of
solutions, does not sustain branching.
I am obviously skirting the central question as to why the 10-adic
solutions to n^2 = n and n^3 = n differ so much in character.
Not being versed in adic theory, I cannot offer an explanation in
terms of adic theory, I could offer an explanation in terms of
modular argument, but it would be too involved for me to do here.
----- Original Message -----
From: "ZAKIRS" <zfseidov at ycariel.yosh.ac.il>
To: "Seqfan at Ext.Jussieu.Fr" <seqfan at ext.jussieu.fr>; "???? ?????? -
?"?/Zakir Seidov Ph.D." <zfseidov at ycariel.yosh.ac.il>
Sent: Monday, September 23, 2002 4:10 AM
Subject: branching in p-morphics
> %C A018247
>
<http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A
> num=A018247>
> Is it known that these two automorphic numbers are distinct? - Zakir F.
> Seidov (seidovzf at yahoo.com), Sep 19 2002
> %C A018248
>
<http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A
> num=A018248>
> Is it known that these two automorphic numbers are distinct? - Zakir F.
> Seidov (seidovzf at yahoo.com), Sep 19 2002
>
> i think i don't understand ( due to my poor english?) this version of my
> comment -
> i didn't ask whether these two numbers are distinct (that is differ from
> each other ?)
> but whether they are unique.
>
> hence my q to seqfans:
> Is it proved that each time, seeking for next digit, one finds only one
> solution?
>
> i mean that in principle it's possible that say 1427th digit may be say 3
> and 8;
> to be more clear - this occurs e.g. in the case of trimorphic numbers:
>
> A033819
> Sequence: 1,4,5,6,9,24,25,49,51,75,76,99,125,249,251,375,376,499,501,
> 624,625,749,751,875,999,1249,3751,4375,4999,5001,5625,6249,
> 8751,9375,9376,9999,18751,31249,40625,49999,50001,59375,
> 68751,81249,90624,90625
> Name: Trimorphic numbers: n^3 ends with n.,
>
> where 3-digit number 751 generates two 4-digit numbers 3751 & 8751,
> and 4-digit number 1249 generates two 5-digit numbers 31249 & 81249, etc.
>
> My q is: is such a branching possible for automorphic numbers? yes or no?
> ken o lo ?
>
> many thanks, zak
>
> ps a quite another point : if there's reference to the classic text why
keep
> in SEQ comment like
> "somebody said me that he'd also found it some times ago".
> it reminds me the old (sorry russian) joke that some renowned expert in
> Pushkin's poetry
> (if someone by any chance forgot - Pushkin is the great poet) yesterday
> discovered
> Pushkin's lyrics - unknown to him before - in... 5th volume of
Pushkin's
> complete works!
>
> pps sorry for long mess
>
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