Subject: -- FASyL to approach very difficult arithmetic problems ;-)
Date: Sat, 17 Oct 1998 17:41:05 +0100
From: Nico Benschop
Org'n: AmSpade (Digital) Research
Newsgrp: sci.math
----------- FASyL : Finite Associative System Logic ---------------
---------------(Semigroups applied to Arithmetic)------------------
It is worthwhile to wonder *why* certain well known problems
are so difficult to solve, for instance in arithmetic:
FLT inequality, Waring Powersum repres'n, Goldbach Primesum Conj,
(3n+1)/2* halting problem, Odd perfect nrs, Twin Prime Conjecture,
Riemann zeta function hypothesis (roots in the complex plane), ...
Is it a coincidence that all of them concern the relation between (.)
and (+), namely the first six with integers, and the last one with
roots (mult've) of a series (add've, with "inverse prime" terms) in
the complex plane. If a good grip on the first six is obtained, the
last one may also yield.
It is not so much "numbers" but "closures" (operations) that should
be considered, and their sub_closures and images (projections).
----------- "Integration" : from Numbers to Systems ----------------
------- On generators, closures, and components (factors) ----------
In this case (.) versus (+), which luckily are strongly connected:
(.) distributes over (+) because (.) is defined by iteration of (+).
So does (^), as iterated (.), distribute over (.). But by Pascal's
triangle (;-) we have (^) NOT distributing over (+),
nor is (^) associative or commutative: a VERY nasty operation.
-- Not for nothing one calls it "ex-ponent": ejecting you out of
the (2-dimensional) plane of arithmetic Z(+,.)
In more general terms: (.) = Endo(+), thus (.) is formed by the
"internal symmetries" = endo-morphisms of (+), in terms of semigroups,
that is: associative algebra of function composition, my pet point of
view (;-) Because it's umbrella allows integration of (.) and (+).
In other words, (.) and (+) form a hierarchy of two semigroups of
which the higher one is formed by the "generalized symmetries"
(endo morphisms) of the lower one. Typical Ring structure...
And doing this in the finite (mod m) takes the 'byte' out of (^).
-- For instance p-th power residues F_k = {n^p mod p^k} form a subgroup
of the group G_k(.) = {n coprime to p} of units mod p^k, with |G_k| =
(p-1)p^{k-1}, and |F_k| = |G_k|/p. A most interesting add've property
of 'core' subgroup A_k(.) of order p-1 (with n^p=n mod p^k) is:
---> each Core subgroup (1 < S <= A_k) sums to zero (mod p^k).
See further http://de.arXiv.org/abs/math.HO/0103051
Now why so difficult about something so simple as finite (discrete
integer) aritmetic? Because apparently something in the approach to
essential problems in Number Theory has been missed,
and by -- Euler's trick:
--> if you cannot solve a *really* difficult problem: generalize it,
in order to see its context - and another way around it for a missed
approach. -- Which in these cases seems to be the point of:
--> studying closures in stead of numbers. <---
Especially: add've properties of mult've closure Z(.) mod m_k, for
properly chosen modulus m_k, followed by induction on k (going to
integers, using the 'carry' beyond the modulus).
All endo-morphisms of a system (semigroup) S, here Z_m(+) = (+1)* mod m
[ addition mod m, forming a cyclic group of order m, generated by 1
under iterating +1: Peano ] are formed by all mappings of a minimal
set of its generators "structure preserving" into the system itself.
This way you get each n in Z_m(+) (n < m) to mean a multiplier by n,
hence an element of Z_m(.) as multiplication mod m.
And that is why |Z_m(+)| = |Z_m(*)| = m: there are precisely m ways
to map generator 1 "structure preserving" (endo morphism) into its own
closure.
The clue in such structure analysis problems is apparently:
------ look for additive properties of the mult've closure, ------
that is: add've analysis of Z(.) mod m.
And: adapt modulus m to the problem at hand:
1. FLT is about adding p-th powers (prime p suffices for the proof
of inequality for integer p-th powers). So take m = p^k (prime p,
k-digit arithmetic), with {n^p} forming a subgroup of Z_m(.)
And use base p for number representation.
After all, some inequality must be shown:
how could that be done without a proper unique representation?
Of course, for integers you want k-->oo, and there is a clinch:
"Hensel lift", see http://www.iae.nl/users/benschop/selmer.htm
and http://www.iae.nl/users/benschop/scimat98.htm
"His_story" on http://www.iae.nl/users/benschop/ferm.htm
2. "Waring" powersum representation problem is more interesting than
FLT, which is about an inequality: not very exciting, is it?-(
"Waring" on integer powersums: n = \sum (n_i)^p , i=1..g(p)
How many p_th powers are required to represent each natural nr?
For p=2 (squares) Fermat & Euler & Lagrange showed already that
each nr is the sum of at most four squares: g(2)=4. Much more, but
not all, is known for larger exponents (# terms grows quite fast).
There is an interesting connection between Fermat and Waring:
FLT's inequality is actually an "anti-closure", since the sum
of two p-th powers is NOT a p-th power! Now this smells clearly
of a good additive generator of the naturals, doesn't it?
(just intuition: anti-closure is the sign of a good generator;-)
And indeed: considering prime power residues n^p mod p^k
it turns out that the sum of at most four of them suffice to cover
all residues mod p^k (for any prime p>2, and any precision k>0).
See http://de.arXiv.org/abs/math.GM/0103083
3. Goldbach's conjecture (each 2n>4 is the sum of two odd primes)
again is an add've property of multiplicative terms (primes are
the mult've generators of the naturals). Clearly the add've
analysis of Z(.) mod m requires here a choice of m that somehow
concentrates the primes in a sub-closure (subsemigroup). This
can be done by m = m_k = \prod p_i (i=1..k) = the product of the
first k primes. Then all primes between p_k and m_k generate sub-
group G1 of Z(.) mod m_k, called its group of "units". And indeed,
each pair of complements in the (Boolean) lattice of idempotents
in Z(.) mod m_k sums to 1 mod m_k, and each even idt +1 is in G1.
So G1 has an additive property: G1 + G1 = E (mod m_k) where E={2n}
is the set of all even residues. So: each even residue (mod m_k)
is the sum of two units (roots of unity): Goldbach for residues.
Going to integers (naturals), is a matter of induction over k.
See http://de.arXiv.org/abs/math.GM/0103091
4. Also the "(3n+1)/2*" problem (of ending in 1 for every seed n)
is an add've statement, with a mult've algorithm:
For odd n: do *3 +1, then divide by the maximum 2^i,
to yield the next odd n, and repeat. -- Conjecture: for each
starting n, the algorithm ends in 1 (the only fixed point?-)
So *what* is so special about 3 as mult've generator?!
There is a clue. Guess what modulus for representation to take?
5. A perfect n (again) is characterized by a special relation between
(*), or rather its inverse (/), and (+). Namely n is "perfect"
---- as the Greeks of old felt it should be, for balance ----
if it eaquals the sum of its divisors (include 1, exclude n).
So 6, 28, 496 are the first three perfect nrs: n=2^(p-1).(2^p -1)
for prime p and (Mersenne) prime (2^p -1). It is not (yet) known
if there are odd perfect nrs. Computers tell us that if they exist
they must be *very* large -- none is found yet.
Again: an add've statement about divisors (mult've_type terms).
6. Twin prime conjecture:
There is an infintude of primepairs differing by 2,
Generalized (relaxed) to: with difference 2^i (for some i>0).
See my recent sci.math reply (15oct98 sci.math) to Russell Easterly.
Just some thoughts on arithmetic (.) and (+), and the taming of (^)
-- which is not associative nor commutative -- by taking first
a generative view of "reflections in the finite" (mod m)...
Ciao, Nico Benschop -- http://www.iae.nl/users/benschop
Dynamic Balance: One is Always Halfway Anyway : 1=AHA
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