Request for information about basis representation
Marc LeBrun
mlb at fxpt.com
Mon Feb 28 22:30:23 CET 2005
>=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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