branching in p-morphics

David Wilson davidwwilson at
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>
To: "Seqfan at Ext.Jussieu.Fr" <seqfan at>; "???? ?????? -
?"?/Zakir Seidov Ph.D." <zfseidov at>
Sent: Monday, September 23, 2002 4:10 AM
Subject: branching in p-morphics

> %C A018247
> num=A018247>
>  Is it known that these two automorphic numbers are distinct? - Zakir F.
> Seidov (seidovzf at, Sep 19 2002
> %C A018248
> num=A018248>
> Is it known that these two automorphic numbers are distinct? - Zakir F.
> Seidov (seidovzf at, 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
> 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
> complete works!
> pps sorry for long mess

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