AW: Re: Thematics Of Order

[Creighton Dement]

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> Date: Fri, 15 Apr 2005 20:14:12 +0200
> Subject: Re: Thematics Of Order
> From: Jon Awbrey <jawbrey at att.net>
> To: Inquiry <inquiry at stderr.org>, Peirce List
> <peirce-l at edsel.tosm.ttu.edu>, SeqFan <seqfan at ext.jussieu.fr>

> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> 
> TOO.  Note 2
> 
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> 
> Re: TOO 1.  http://stderr.org/pipermail/inquiry/2005-April/002541.html
> In: TOO.   
> http://stderr.org/pipermail/inquiry/2005-April/thread.html#2541 
> The idea of "reflective order closure" (ROC),
> whereby a "statement about the order" (SATO)
> becomes a "statement in the order" (SITO),
> arose while exploring the lambda point
> between algebra, geometry, and logic,
> where algebra includes things like
> group theory and number theory,
> and where geometry includes
> things like graph theory.
> 
> Many of the things that puzzle us the most in number theory
> have to do with our lack of knowledge about the connections
> between additive properties and multiplicative properties
> of A000027, also known as the sequence of natural numbers.
> 
> For instance, if we take a very nice representation of natural numbers
> in multiplcative terms, namely, their primes factorization
> expressions, and try to add two numbers in this form, then we have no
> direct way of doing this that does not "cheat" by resorting to their
> additive forms.
> In particular, we do not know how to add 1 to a primes factorization
> and get the primes factorization of its successor in any direct way.

I shall risk asking a real layman's question here:
Assuming the phrase "direct way" can be given a precise mathematical
defition, what is known about the set 
{ (c(n)) | c is a (positive) integer sequence; if n has known prime
factorization, then the prime factorization of n + c(n) can be
calculated in a direct way. } . This set immediately seems to be
infinite, as any c(n) =  (p-1)n
for prime p would be a member.  Your statement, above, would then be
equivalent  to saying that c(n) = 1 for all n is not a member of the
set.  Are there nontrivial examples of elements which are in the set
(say, which rely on theorems)?

Sincerely, 
Creighton 

> And this amounts to saying that we do not know how to order the primes
> factorizations in their usual additive order simply by
> looking at their multiplcative representations.
> 
> Reflections like these eventually bring us to the question:
> 
> How much of the additive, linear, or total order in the
> sequence of natural numbers is "purely mutiplicative",
> that is, how much of the order can be determined
> solely by inspection of primes factorizations?
> 
> Jon Awbrey
> 
> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> inquiry e-lab: http://stderr.org/pipermail/inquiry/
> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> 
> 
>