Primal Mask Ops, AKA Ripple Caryatids

[Jon Awbrey]

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PMO.  Note 1

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Re: (A XOR B) + 2 (A AND B) and Ripple Carry Adders

Antti, SeqFans ...

Another antic idea that comes up in (free) association
with the consideration of bit mask operations is this:

Consider the finite sequence of primes' factorization exponents, zeros included,
that we get for each multiplicatively natural number m in M = {1, 2, 3, ...}.
I used to call this the "primal logarithm" or "vector logarithm", but I rechnen
that everybody probably has their own pet name for it.  We can also think about
the associated multiset of prime tokens, contemplating logical or set-theoretic
operations, and visualizing venn diagrams for these logocal operations.

For concreteness:

360 = 2^3 3^2 5^1 = <3 2 1> = {2 2 2 3 3 5}

999 = 2^0 3^3 5^0 7^0 11^0 13^0 17^0 19^0 23^0 29^0 31^0 37^1
    = <0 3 0 0 0 0 0 0 0 0 0 1>
    = {3 3 3 37}

These representations make multiplication very easy --
we just add the primal logs, place by place primewise --
But is there any algebra or logic associated with "+"?
To a 'very rough' first approximation, adding numbers
is analogous to logical equivalence, in other words,
to the complement of the symmetric difference, since
we know that the sum is divisible by the GCD, and
has no primes in the sym diff part of the p-logs.
But there may be extra primes in the AND part,
and extra primes in the NNOR part.

For example:

 360 = <3 2 1>                   = {2 2 2 3 3 5}
 999 = <0 3 0 0 0 0 0 0 0 0 0 1> = {3 3 3 37}
-----------------------------------------------
1359 = <0 2 0 ... 0 1_36>        = {3 3 151}

Here 1_36 means 1 in the 36^th place.

Of course, some of the hardest problems in number theory
are disguised beneath this innocuous exterior, but it's
still sum fun to think about things this way.

Jon Awbrey

P.S.  I am copying this thread to a list archive,
as I have trouble remembering what I was thinking
about last week without a list and a web searcher,
so anybody who responds please let me know if you
don't want your replies e-graven for posterity.

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