Request for information about basis representation

[Marc LeBrun]

 >=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?)




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