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Nico F. Benschop

Essential Motivation (C.Shannon)

John Glenn (77), Shuttle #92: Discovery - 7nov1998 @ Kennedy Space Center... Proficiat!

Charles Lindbergh (25), Spirit of St.Louis - 21may1927 @ Le Bourget, Paris... touch down!  

Finnish translation

. . Home email: . . AmSpade Research

Edouard Manet (1882): Chrystal vase + clematis

                            Some WWWisdom on Logic, FLT, Goldbach, Automata
(beware: with sci.math Core ;-) and other Digital Math.

"State Machine decomposition by Right-congruences" . . . . (Asymmetric time)

Cantor's uncountable strings A* over alphabet of |A| > 1 letters (sci.math feb'02)

A simple FLT proof _is_ possible : via the cubic roots of 1 mod p^k.

. . . . . . Goldbach elementary proof

. . . . . . FLT elementary proof (p169-184)

 

 


- My new BOOK - preface.htm , Cover
- - - The Associative Structure of State Machines
-- an Associative Algebra Approach to Logic, Arithmetic and Automata

Order  (19.50 EU) at publisher abc.nl (Amsterdam, ISBN-search ‘9789491030031’)

See ACM Computing review, and Zentralblatt.Math review ,
and
Wikibooks.org . . and Wikipedia.org

If stuck at Closure: use the Carry . . . Residue-and-Carry method to prove FLT and Goldbach

Symmetric and Planar Logic Synthesis . . . Each boolean function BF_n for n<5 is planar

 

 

Homepage2 on Astro-physics (redshift, foton-decay, ether):
. . . . . Including home page Mike Lewis for up-to-date material (under ref [2])

Truth depends on Axioms ::: Axiom = Self-evident = a matter of Opinion = Belief.
. . . Ergo : Truth is Belief . . . (thus subjective, after all;-)

On New Ideas in Science (and resistence to them), by Thomas Gold (1989).

Paul Marmet: "...explanations which are compatible with Newtonian and semi-classical physics.
We apply the principle of causality and avoid models requiring a probabilistic existence of matter
and non-locality: with a stable (non-expanding) Universe, different from modern physics."

BigBang is wrong, by John Kierein

Hubble redshift as non-Doppler effect:
Photon decay ('tired light') in space by Inelastic collision of photons with Molecular Hydrogen.

- Some of my papers on Arxiv (at front.math.ucdavis.edu)

"Commonsense-in-Science Manifesto"
=====================================

1. . . A wave phenomenon requires a medium,
So : . . . . a wave cannot propagate in vacuüm.

2 . . . . EM waves (light) propagate in space,
So . . Space is not vacuüm (say 'ether' medium).

3. . . Perpetuüm mobile (lossfree) processes do not exist,
So : . . . . . Wave propagation (light travel) is not lossless
And if linear loss : redshift proportional to distance travelled.

4. . . Speed of wave propagation depends on medium density,
And : . . . Density varies with Gravity field around heavy body. [!]
So : . . . Interstellar space is filled with inhomogeneous medium,
So . . . Lightspeed is not constant, and light bends around a star.
Ref:
- Redshift by Photon Decay

5. . . With matter (electron) as closed wave (closed photon) [*]:
It makes sense that rotating matter (Earth) drags ether along.
So: . . This adherence prevents ether-motion measured along Earth surface.
(Michelson_Morley's null-result is no surprise, but their
'conclusion' (no ether) _is_ surprising, based on insufficient
grounds: neglecting the alternative of ether-adherence to Earth)

6. [!] . . If ether density around heavy body decreases as 1/r
And : . . . Gravity is defined as its gradient
Then: . . . Gravity decreases as -1/(r^2) ---> Newton's gravity Law [!]

7. The constancy of lightspeed c holds for all frequencies with accuracy 10^{-21}.
"The speed of light is independent of frequency within a factor of 6x10-21" (B.Schaefer)
. . This points to the existence of a medium that determines speed c, since photons
. . of widely different frequencies could in vacuüm travel at various speeds as
. . determined at their generation, and maintain those various speeds forever.
So: Einsteins axiom of constant speed c induces a medium for light propagation.

8. Lightspeed c is the natural wave propagation speed in the ether medium.
. . Nothing compels it to be the absolute maximum speed (as assumed by Einstein).
. . Similar to the speed of sound in air: a projectile with enough energy can break
. . though the sound barrier (e.g. a jet airplane). So a neutrino with enough energy can
. . break through the lightspeed barrier, possibly with a V-cone shockwave.
. . There is no connection with causality problems, time reversal, twin paradox, etc.

Support Wikipedia

[*] Is the electron a photon with toroidal topology? [*]
by Martin van der Mark, J.G. Williamson (.pdf - 1997)
'All is EM field' - including elementary 'particles'
(
donut structure of electron)

Quasars, Big-Bang & Cosmological constant by dr. Paul Marmet
From: IEEE Transactions on Plasma Science, V20, N6, pp958-964, 1992.

"Big Bang, Black Holes, and Common Sense" by David Pratt

Redshift History by R. Gray and J. Dunning-Davies (Hull Univ UK)

"A new non-Doppler Redshift" by dr. Paul Marmet.

On Fermat's marginal note, a suggestion N.F.Benschop (NMC-33, Apr.1998)

My father's diary on his Birma Railroad experience 1942-'45 (in Dutch).
Buy hardcover copy (Lulu.com) 75 pgs or Softcover.


Curiosity and Search . . . . . "An open mind is a joy forever "

. Inertia . . . . . . . . . . . . . . . . " Climbing Mount Prejudice "

. . Generalize . . . . . . . . "If a problem is too hard, generalize it "

. . . Simplify . . . . . " We have to fight uphill to rediscover the obvious "

. . . . Coincidence . . . . . . . . . " Life is fighting entropy "

My CV . . . Work . . . Play . . . ff . . finite fun . . (reflections in the finite)*

. . . . . . Intuition : . . Even if you don't see it, you still can smell it !

. . . . . . Bamboe . . . Chinese brush painting

Remembering: - (NB: 28-july-2002)
Claude Shannon

Fermat, Waring, Goldbach -:- view Arithmetic as Function composition (semigroups) :
. . . Additive Number Theory : hard problems, solved via semigroups Z(.) mod m_k , and . . .

"Additive structure of the Group of Units mod p^k ,
with Core and Carry concepts for extension to Integers"

- - - Triplet structure, and direct FLT proof (16 pgs), published Nov.2005 in:
AMUC - Acta Mathematica Univ. Bratislava

Péano- and Cantor- sets and their generation: sequential dimension


The sawtooth of Evolution :
. . Extremes increase _ - + * ^ and then collapse \___ . . &c (with positive Carry ?-)

Child's arithmetic ( = exponential ;-( . . . Stable economic growth of 3 % per year,
. . . then *(1.03)^n yields economy *10 ... *100 ... *1000 after n = 78 ... 156 ... 234 years !

NB: . . . One Generation = 26 yrs ==> Your grandchildren live 10 x dense than you . . .

Humor / Relegion
"If God created us in his own image, we have more than reciprocated."
Voltaire (1694-1778).


Links (en Rechts;-) + Discussions . . . my 8 papers at : de.arXiv.org

Among others :
- Finite Semigroups of Constant Rank, and the five Basic State Machine types.
- On Log-arithmetic, Fermat's Small Theorem, and the powers of 3 (mod 2^k).
- Extending Fermat's Small Thm to r^{p-1} mod p^3, for divisors r | p +/- 1.


My cup-a-soup :

^ * + - . : . - + * ^
Once upon a time there were three kinds of math-objects : Sets, Numbers, Functions (between Sets)
Set |S|=n has 2n subsets, n! permutations, nn transforms (say n=5 : 2.2.2.2.2 < 1.2.3.4.5 < 5.5.5.5.5)
Sets (logic) compose by intersection( /\ ) and union( v ) which are both idempotent : a/\a=a, a\/a=a.
Numbers you can add(+) and multiply(.) which both are commutative: a+b=b+a, a.b=b.a
Functions (seq_logic) compose by sequencing ( ) which is associative: a(bc)=(ab)c.
Including logic and arithmetic, which are also associative, we get Semigroups,
making a triple sandwich of logic < arithmetic < function-composition,
and the 3 algebra's are idempotent < commutative < associative.
Like order and equivalence: reflexive | symmetric | transitive.
The two sides of the same coin : objects form semantics,
sequential behaviour of composition rules is the syntax.
- Exponentiation (^) falls out of the boat for misbehaving -
It is not idt, cmt, assoc; ---- It distributes over (.) but not over (+)
So it becomes really interesting when you combine (+) and (^) as in x^p + y^p = z^p.
with no integer solution for p>2: the powersum yields no p-th power, but what then ? [6]
++ It turns out that p-th power residues mod p^k are the best additive residue base ++
Symmetries -n and 1/n mod p^k, with n+1= -1/n yield FST --> FLT [3], and other stu ff.



M S D : More specific designs: . Powersums [3 1 5 6 7] . . 3* code [21] . . FSM seq_logic synthesis [2]:

[-4] Internal State of Computer Science (ICS73.pdf)
. . . . . . . My original motivation for Digital Network Theory <=> Finite Semigroups
[-3]
Re: unify all branches in math - and some math history (sci.math 19feb02)
[-2]
Re: Problem with Cantor's diagonal argument
. . . Cantor's Diagonals generate powerset 2^N by permutations of w x w table rows (sci.math 15feb02)
[-1]
Fermat's Anti-Closure n^p (+) as Generator (.htm) . . . . . . . . . . . . news:sci.math - 28nov98
[0]
Function composition of symmetries -n, 1/n for FLT (.htm) . . . . . . . . . . . . sci.math - apr98
[1]
Triplets and Symmetries of Arithmetic mod p^k (.pdf) . . . Semigroups, Arithmetic (Fermat Last Thm FLT)
[2]
On the structure of Constant-Rank State Machines (.pdf) . . Sgrps and Sequential Logic Synthesis
[3]
Fermat and the Cubic roots of 1 mod p^k (.html) . . . . . . . .Breaking the Hensel lift . (1 page clue)
[4]
On the triplet structure of arithmetic mod p^k,....
[5]
On Fermats' marginal note: a suggestion (.pdf) . . . Cubic roots: a^3=1 mod p^k (marg-abs.htm)
. . . .
Quotes E.T.Bell (.htm) : . . . . . . . . . . . . On abstraction, specialisation and divergence.
[6]
On powersums representing residues mod p^k(.pdf) . . . . " Waring-for-residues " (nw-abstr.htm)
. . . . . -------- Each residue mod p^k is the sum of at most four p-th power residues --------
[7]
Arithmetic with squarefree modulus, its lattice of groups,
. . . a primesieve, and Goldbach's conjecture(.pdf) . . . Goldbach proof (old version 2002):
[8]
-- Fermat, Waring, Goldbach -- (.htm) . . . . . . . . . . . . . . . . . . via Semigroups Z(.) mod m_k
[9]
One more crank (.htm) sci.math - may98 . . . . . . . Hasse-Minkowski thm and Selmer's group
[10]
How diagonal is Cantor's diagonal ? (.htm) . . . A Bootstrap process generating the reals.
. . . . . . . . . . .
sci.math 5jun98 . . 6jun98
[11]
Semigroups and FLT (.htm) . . Z(.,+) mod m . . . . . . Function composition linking (+) and (^)
[12]
New Math (base p) applied to FLT (.htm) . . . . . . . . (Fermat?) computing a^3=1 mod 7^2.
[13]
Re: Orthogonality (.htm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sci.math - 3nov98
[14]
Re: Primitive roots of Unity (.htm) . . . . . . . . . . . . . . . . . . . . . . . . sci.math - 3nov98
[15]
-- Stumbling over Aladin's lamp .. sci.math 13nov98 . . . . . . . . . . sci.math 10nov98
[15']
A simple direct FLT proof after all ? (despite the Hensel lift) . . . . sci.math - 1aug00
[16]
Fermat's anti-closure n^p(+) as generator / BOOA constructor . . . sci.math - 4dec98
[17]
Re: Finite versus Infinite (a matter of taste) . . . . . . . . . . . . . . . . . . sci.math - 16feb99
[18]
Re: Why the attack on Cantor's Theory? . . from Cardinals to Types . . . sci.math 27feb99
[19]
Easter primes do not exist: 2^p != 2 mod p^3 (prime p) sci.math 12apr98, and :
[19b]
Divisors r of p-1 (resp. p+1) have distinct r^{p-1} mod p^3 . . Carry-extend Fermat's Small Thm. (.pdf)
[20]
The role of prejudice in the exact sciences . . . a non-discussion on nmbrthry.list (14mar99)
[21]
My +/- 3*2* Binary Multiplier . . . . US patent 59.23.888 (13 july 1999):
. . . 3 is semi-primitive root of 1 mod 2^k, Log-arithmetic (dual base 2,3) ..
(.pdf)
[22]
Sequential dimension of finite and infinite closures: N vs 2^N . . . sci.math 7oct99
[23]
ISM : Integer State Machines . . . . . . M(Z,A) state set Z, inputs A={+1, -1,swap(2)}
[24]
Discovering the Triplets in Z(.) mod p^k . . (a+1)b=(b+1)c=(c+1)a = -1, abc = 1 (mod p^k)
[25]
On FSM-decomposition and Semigroups (= sequential closure): part I . . . . . part II
[26]
On Reals (as un-real concept) : . . One can specify only countably many reals anyway!

__ L S D : Less simplistic directions [*.html] : . . . a wider view - beyond Logic and Arithmetic :

[a] . Cubic roots of 1 . . . . . . . . . . . . . . . . . Triplets /3/ and Symmetry /2/ of FLT mod p^k roots
[b] .
Residue and Carry . . . . . . . . . . . . . . . . . for Closure and Generation ( confirm and explore)
[c] .
A day with Fermat in Toulouse . . . . . . Did he find the cubic root solution of FLT mod 7^2 ?
[d] .
I-Ching (symbol 59 and text) . . . . . . The Solution . (of a stuck-at situation: whip-up water)
[e] .
Bibliotheca Philosophica Hermetica (Amsterdam)
. . . " Newton wrote more on alchemy than on any other subject" (we lost something? Why vs. How)
[f] .
I-Ching : 8 basic icons , Yi-Jing info . . Chinese philosophy: Balanced Change, Cyclic Time
[g] .
On Dynamic Balance and Bootstrap . . . On a balanced psychology (Maslow and Spangler)
[h] .
Weighing Cantor's Evidence . . . . . . . . . . . . . . . On math, EE-practice and real infinity
[i] .
Red-shift of fotons by decay: Ether after all? . . . . . . . . . . On lossy light propagation.
[j] .
Strange dominance of math in world model . . . . . . . . . Maslow's "hammer and nails"
[k] .
Re: Pi is no piece of Pie . . . . . . . . . . . . . Finite atomism vs Infinite partition (sci.math)
[l] .
On controversies . . . . . . . . Finite/Infinite, Discrete/Continuous, Tradition_/Modern_units
[m] .
Something Rotten . . . . at the Core of Science? (Trends in Pharmacological Sciences)
[n] .
On Digital physics (Fredkin) . . . and Cellular automata (Steven Wolfram)
[o] Bush' Mission to Mars (TIME 23feb04): " Sending humans to Mars sounds
. . . fantastic, like something to strive for. But the problem is that the mission will
. . . just give us another world to mess up. Untill we have a complete change of
. . . attitude and behaviour, let's stay away from other planets!" (L.Fourie, S-Africa)
---- "Why should we spend vast amounts of money to reach Mars when the chance is very
---- high that the mission does not provide anything to improve our lot here on Earth?
---- Why not allocate those billions for "home improvement"? How about curing cancer,
---- aids and other diseases; cleaning our water, air and soil; feeding hungry nations.
---- [NB: Or making the Sahara fertile. A few hundred billion dollars/euros can go a long way!-]
---- Why confront the hazards of outer space when we should be fighting homelessness,
---- poverty and unemployment on Earth? (C.Brunish, Oregan, USA)

---------------- if stuck@closure (mod ...) : use the Carry ! ----------------

--- from Numbers {n+p, n.p, n^p} to Sytems Z(+)(.) and (^) mod m ---
--------------
FASYL . : . Finite Associative Systems Logic ---------------

All updates
2000 jan12[restored 1,5,6,7] jan23[l] 25mar[j] 11jun[1: lem2.2,3.1] 2aug[15',24,25] 5aug[0: sgrps] 3sep[6:thm3.1] 9sep[i] -- 2001: dec['Links'] -- 2002: 15feb[-2] 2sep[Photon decay] 30oct[19b] -- 2003: 18feb[26] 16apr[-4] 19may[27][m] 21may[mycv: my book] 27aug[n] 1sep[i: Maxwell demon ref] 15oct[10: seqdim intro] 4nov[play, CS-preprints] 20dec[links] -- 2004: 25jan[intuition.ppt] 26feb[o: Mars] 27sep[DNT Book]


NB : At a philosophical & common-sense level, Godel's result came surpisingly late,
. . . just saying that every theory (as transitive closure of a finite set of axioms,
. . . using logic/causality as generating principle) has its context.
So : TRUTH is relative to your assumptions (axioms), and math in particular
. . . . . . . does not yield anything really new beyond their transitive closure !


[V] . Campaign : for the Cubic roots of Unity, breaking Hensel's lift (re: FLT), and
to promote finite transformation Semigroups and functions for Logic, Arithmetic, FSM's [2].
--- If you like this direct approach [3] to FLT (case_1): tell a friend (or even me ;-)
--- And if you do not like it : please tell me why;
. . . certainly if you can disprove the EDS argument [3] for the cubic root solution.

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