a tad more on OR-numbrals

[Marc LeBrun]

 >=Jens Voss

 > In fact, it is not hard to see that [2] is the only OR-numbral
 > prime number.

Very nice argument!  This is the kind of development of "numbral theory" I 
am hoping to see.  Each concrete system    possesses a rich (unexplored!) 
special analog theory, to say nothing of the general questions about what 
systems are isomorphic to eachother or to other known algebraic structures...

 > Recall that a number is defined to be _prime_ if whenever it divides
 > a product, it must also divide (at least) one of the factors.

Not to quibble, but I'm not completely comfortable just accepting this as 
the definition of prime.  Is there some sort of "classical" development 
that necessarily begins with this?

I don't imagine, for instance, that the ancient Greeks would have chosen it 
over the seemingly more natural (generalized) definition as "having no 
non-unit proper divisors".  (Though of course these mutually imply 
eachother for ordinary numbers).

This sort of "splitting" is typical of numbrals, and it's not clear to me 
which end of these elephants we should start with in building up analog 
theories.  Maybe both: my preference is to pick definitions that lead to 
richer systems, even if some of the theorems for regular numbers have to be 
modified or abandoned.  But any guidance or suggestions are welcome.

For example not being a UFD breaks the normal Euclidean algorithm, yet the 
concept of a GCD as "largest common divisor" still seems perfectly 
natural.  We just can't deduce all the same consequences.  Similarly, every 
XOR numbral can be represented by infinitely many positive sums, but if we 
restrict it to "proper addends" we can meaningfully define XOR-partitions, etc.

By the way, the sequence giving the number of OR-partitions actually made 
me laugh out loud when I first saw it.  It looked so absurd (compared, eg, 
to the XOR-partitions) that I was sure my program was whacko!  But it in 
fact turned out to be based on a deeply serious sequence--in the EIS thanks 
to Michael Somos--that's been whipped in a blender like mathematical 
margarita mix by Zippy the Pinhead.  (Is there an alt.humor.sequences?)

Anyway, thanks for the interesting analysis!

Happy New Year!