Subject: Re: Benschop on arXiv
   Date: Thu, 17 Jan 2002 
   From: Nico Benschop 
Newsgrp: sci.math

ayatollah potassium wrote:
>
> Nico Benschop wrote:
>   
> > [...] just keep ignoring that approach, keep supporting RC in 
> > his tirades against me (see above in this thread) and you are
> > certain to make no progress towards insight in these basic /
> > elementary aspects of integer arithmetic,
> >                     requiring techniques from finite semigroups.
> > -- "A closed mind is a block forever";-(      ^^^^^^^^^^^^^^^^^
>
> blast from the past:
>  "a little bit of physics would be NO idleness in mathematics".

Agreed, I see this as: a bit of intuition brings you a long way.

Hard problems in arithmetic ('additive' nr-th as in Fermat, Goldbach)
require a special tool beyond arithmetic. I suggest:
   Generalize commutative associative arithmetic (+,.) to associative
   function composition, of course adjusted to the problem at hand.
E.g. the two operations (+) and (.) are both assoc & cmt, while (^)
is NEITHER, and each has a 'symmetry', viz. automorphism of order 2,
to be viewed as *functions* :
  complement C(n) = -n,  and Inverse I(n) = n^{-1} , which commute:
CI(n)=IC(n), while the Peano successor function S(n)=n+1 does not
commute with C(n) or I(n).

And there are four distinct compositions of these three fundamental
functions: SCI(n), ICS(n), ISC(n), CSI(n)  -- (watch: n \neq 0,1,-1)
   for instance SCI(n)= -1/(n+1) .. in left-to-right notation
   (Clifford/Preston book on Semigroups, AMS survey #7, 1961).
With the remarkable(?) property that all have period = 3 if iterated:
for instance  SCI(SCI(SCI(n))) = n = E(n) : the identity function for
all n \neq 0,1,-1.

Such function approach to the symmetries (= 'structure') of finite
arithmetic offers a simple way to go further, where known arithmetic
methods fail. That is what I mean with: if a problem is REALLY hard,
--> generalize it, to find a way around it, viewing with other eyes.
Note: NO constructive response to any of this occurred
      in the past 7 years of presentations & submissions.
In the above sense the Fermat and Goldbach problems are not purely
arithmetic, but require additive analysis of multiplicative
Z(.) mod m_k .. by functional (semigroup) tools, and for integers:
re-introduce the 'carry' which was ignored in residue arithmetic,
for easy closure sake... (Gauss, 1801).

-- NB - http://home.iae.nl/users/benschop/fewago.htm