Request for information about basis representation

[Andrew Plewe]

Regarding your last question, I'd like to determine the properties of the factor base for a "rebasing" sequence and whether or not
that factor base follows any particular rules, is random, etc., and not just for 2[n]q but  other bases as well (say, 7[n]q or
15[n]q).  Another question I have is if prime bases produce sets with different properties than composite bases.  

To make things more interesting, what sorts of things happen if we perform various arithmetic operations using base q logic on n and
its counterpart in the rebasing sequence?  We can start with straightforward logical circuits such as half-adders implemented in
base q logic (the binary AND generalizes to floor((a+b)/n) and XOR to a+b = XOR mod n, for some digits a and b in base n), but since
the possibilites for different truth tables multiply rapidly as q increases, the possibilites for doing other strange sorts of
operations multiply rapidly as well.

Quick note; Edwin Clark off-list pointed out a basic mistake I made in my original email -- it's incorrect to state {1, 0, 0, 0, 1,
1} is a set of coefficients, rather it's a sequence of coefficients and should be written (1, 0, 0, 0, 1, 1).

	-Andrew Plewe-

-----Original Message-----
From: Marc LeBrun [mailto:mlb at fxpt.com] 
Sent: Monday, February 28, 2005 1:30 PM
To: Sequence Fans
Subject: Re: Request for information about basis representation

 >=Andrew Plewe <aplewe at sbcglobal.net>
 > I'm interested in the algebraic properties of the set of numbers whose coefficients S are fixed but whose base varies from two to
infinity. In the case of 35, this set is {35, 247, 1029...}

I'd be fascinated to learn of any results you uncover.

Alas I don't have any references for you, but indulge me an advertisement for a proposed notation:

What you are describing is an instance of what I suggest be called "rebasing", notated b[n]q, and interpreted as "replace b with q
in the expansion of n".

Your examples would thus be written 2[35]2, 2[35]3, 2[35]4...

Many interesting sequences (eg 2[n]4 Moser-deBruijn, A000695) and operations (eg 10[n](1/10) ~digit reverse, shifted) are nicely
expressible this way.

The notation seems congenial, such as q[n]b as inverse for b[n]q, etc.

It's also natural to generalize the idea of "basis" so as to cover the likes of F[n]2, the so-called "fibbinary" numbers (A003714),
and provide standard ready-made images of entities obeying other arithmetics, say like
GF2[n]2 (eg primes = A014580).

Of course if the polynomial b[n](x) factors then b[n]q will be composite for all q, but how far this sort of analysis might lead is
uncertain (eg even if the polynomial doesn't factor, what might divisors of b[n]q tell us about those of b[n]r?)