It is about the use of function composition (semigroup theory) as
binding principle for the three main levels of functions applied in
computer engineering : State-machines, Arithmetic and (Boolean)
logic, corresponding to a hi�rarchy of associative algebra's :
non-commutative, commutative, and idempotent,
respectively : a(bc)=(ab)c=abc , ab=ba , aa=a.
In Chapter 2 the five basic state machines are derived, and
the decomposition of permutation machines with a non-trivial
simple group as closure is in Ch.3 (e.g. group A5 of order 60
as coupled network of cyclic groups of order 2 [twice] , 3, 5).
Ch.4 discusses the general decompositon of finite semigroups /
statemachines, as (possibly coupled) network of the five basic
statemachine types, including the two non-commutative
memory types: branch- and set/reset machines.
'Planar Boolean Logic' (Ch.5) is defined as practically related
to symmetric Boolean functions, with a proof that all BF_n of
n<5 inputs are planar. Also various forms of fault tolerant logic
designs are treated and compared.
Moreover, proofs of FLT and Goldbach's Conjecture (Ch.8, 9)
are given, both using a 'residue-and-carry' method (with proper
choice of modulus), as well as a result of "Waring for residues"
(Ch.10) for prime power moduli: each residue mod p^k [k>0] is
the sum of at most 4 p-th power residues.
Log-arithmetic (Ch.11) over double base 2 and 3 is discussed ,
using : each odd residue mod 2^k is a unique signed power of 3,
as well as over single base 2 (a 32-bit VLSI implementation as
an Euro 'Esprit' project).
dr. Nico F. Benschop, Geldrop (NL) -- Amspade Research ----
[mod note. Springer have clearly published this book by Benschop.
Furthermore, this book claims to give elementary proofs of FLT
and of Goldbach's conjecture. Benschop has posted proofs of
such things in the past---for example chapter 8 of this book
appears to the moderator to be largely the same as the proof
of Goldbach's conjecture announced at
in 2001. As far as the moderator knows, the mathematical community
has not yet accepted the proof described in this paper. Furthermore, viewers
with access to math. reviews might want to see review number MR1831809, which
appears to the moderator to pertain to another chapter of this
book. The moderator wants to make it clear that acceptance of
this sci.math.research post is nothing more than *acknowledgement of the
statement that Springer has published the book*, which the moderator
believes to be of independent interest, and does not imply that
the moderator has read, or believes, any of the stronger claims
made in the book.]
At �81.50 for 180 pages I cannot see this being a bestseller.
I see that this is published by the computer science division
rather than the mathematics division of Springer. I wonder
if the manuscript was reviewed by any competent mathematicians.
> Moreover, proofs of FLT and Goldbach's Conjecture (Ch.8, 9)
> are given, both using a 'residue-and-carry' method (with proper
> choice of modulus), as well as a result of "Waring for residues"
> (Ch.10) for prime power moduli: each residue mod p^k [k>0] is
> the sum of at most 4 p-th power residues.
There we are. A decade or so ago when both Mr Benschop
and I were active in sci.math, Mr Benschop had a reputation
as one of the group's leading "cranks", promulgating his bogus
"proofs" of Fermat's last theorem and Goldbach's conjecture.
I, and others, kept refuting these arguments, at least those
which were clear enough to be refuted. I would be sceptical
whether there is any value in this new book, but I'm not paying
to find out.
A copy of the bibliography and index of the book can be
downloaded from the web link above. I note that one of the papers
cited was Benschop's 2000 paper in the journal "Computers and
Mathematics with Applications". In this paper Mr Benschop
claimed to prove that there was no prime p with 2^p = 2 (mod p^3).
However the alleged proof was bogus and was refuted to me in
a letter to the editor of the journal. Those with access to
MathSciNet can consult
to see the saga.
But while the book cites Benschop's paper it does not cite my
refutation. I consider this to be both dishonest and cowardly
of Mr Benschop.
[mod note: I think Benschop and Chapman have had previous disagreements
about Benschop's claimed results before. I don't want this thread to
turn into a s.m.r. flame war so, now I've let both sides have "one turn"
I've set followups to sci.math; s.m.r. is definitely no place for a debate about
whether Benschop has proved Goldbach or not. ]