Thematics Of Order -- Discussion

[Jon Awbrey]

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TOO.  Discussion Note 2

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CD = Creighton Dement

Re: TOO 2.  http://stderr.org/pipermail/inquiry/2005-April/002545.html
In: TOO.    http://stderr.org/pipermail/inquiry/2005-April/thread.html#2541

CD: I shall risk asking a real layman's question here:  Assuming the phrase
    "direct way" can be given a precise mathematical definition, 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)?

Creighton,

I will have to rely on casual intuitions for the moment,
but maybe I can clarify the starting situation a little.
We are given two numbers represented in terms of their
primes factorizations, either lists of exponents that
terminate with all zeroes after finite intervals, or
finite functions from the indices of the actually
occurring primes to their non-zero exponents.

Now, we can always multiply things out and get
the usual decimal representation of each number,
add them together in that representation, and then
completely factor the sum into its multiplicative
representation, but that would be a clear example
of indirection, since it has to detour through
the additive representation of the addends.

So there's clearly some sort of computable morphism
between the two representations, but the question
is more about the "naturalness" of the analogy,
and I'm afraid that's always going to be a bit
tough to formalize, though not beyond hope.

Jon Awbrey

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