h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Back from holidays .. and from waiting for you to makeup your minds.
By the feedback I got from you foulks, this appears indeed to be the wrong site to draw attention to this simple residue-and-carry approach to FLT - with some historic circumstances starting with an extension of FST (Fermat discovered his small theorem 'FST' about 1640 when he made his well known remark, on a marvelous FLT proof, in the margin of his Bachet edition of Diophantus "Arithmetic").
I won't bother your 10-year-olds with this. So go ahead and remove it from the Peer Review list. -- Best regards, NB.

PS:
For those math-buffs among you who are interested in the full proof,
see online http://pc2.iam.fmph.uniba.sk/amuc/_vol74n2.html
(pp 169-184, Nov'05 issue of the Acta Mathematica, Univ. Bratislava,
who have special expertise on the application of semigroups to arithmetic,
due to prof. Stefan Schwarz who initiated the Semigroup Forum journal in the 1960's)

You write: "It also appears, at one point, that you use 7^2=42, which would be wrong if interpreted
as 'seven to the second power' as many notational forms would have it. 7^2 would, of course, equal 49."

Can you point where precisely? Either it is a typo, or you confuse base 7 with decimal notation.
The cubic roots of unity (mod 7^2) are in fact 42, 24, 01 (in base 7 code), crucial in FLT proof. -- NFB

I've been out of the loop for quite a while.

More on the "human equation", that thrill of discovery you felt when you made the discovery, how did you feel? Communicating emotional states is in many ways a more difficult task than communicating a cognitive truth.

It is precisely this that makes for literary appeal and it takes patience, application and practice. Sometimes I personally go through several hundred brainstorm variations on a theme before I select a few that work for specialized purposes.

Hi ITIWBS,

Fascinating name. Does it mean anything?

You write you're interested: "More on the 'human equation', that thrill of discovery you felt when you made the discovery, how did you feel?"

There are some moments in one's carreer (mine was 32 years in digital electronics/math research) that one does not forget. One of them was, when analysing a log-arithmic way of multiplying (to reduce MPY to ADD which is much simpler in hardware) in k-digit residue arithmetic base p (prime), that two p-th powers *can* add to another p-th power.
(but for fairly obvious reasons, related to Fermat's Small Thm, this cannot be extended to integers).

This occurs for the cubic roots of unity: 1 mod p^k (exponent k>1), when prime p has a value sothat 3 divides p-1 (otherwise there is no cubic root r with r^3 = 1 mod p^2). Fermat discovered this I think, just after his small theorem FST: n^p = n mod p for each n coprime to p (meaning no common divisors) around 1640. This was no doubt the basis of his excitement about the impossibility of extending Pythagoras' theorem for quadratics to higher powers (you know from early school: a^2 + b^2 = c^2 can hold for some integers a,b,c : the link between arithmetic and geometry! via the right angle involved).

As you, I am interested in history, in order to learn how we got into the mess we are in now -- possibly find out why certain developments took the direction they took. Usually it is best to read the originators of some innovation, not the followers! My favourite math history book is "Development of mathematics" by E.T.Bell (McGraw-Hill). He is not popular among mathematicians because he is quite critical of that/his profession! But I love it.

Back to your question 'what did I feel' when discovering some essence: I was thrilled, and I wanted to get out and share with anyone interested. But even at my Lab (Philips Research) there was no-one to share my excitement - by lack of colleagues of the same interests. So I spent several years discussing on the sci.math forum, which I eventually left by lack of interest: if frequently the question is aroused what Fermat could possibly be so enthusiastic about (re his 'wonderful proof' remark on FLT) and I suggest something pertinent (see my entry) .. and then no-one replies, or some are even antagonistic, then I don't understand people's motivations anymore.

A full proof along this cubic-root-of-1 line was published Nov.2005 (see link above), and I mentioned it in sci.math six times (each 7-th of the month from oct'06 -- mar'07). Resulting in only one silly remark by some nutty (or frustrated?) French mathematician ("boycot that University") .. then, as we say in Holland: 'breekt mijn klomp' (klomp = wooden shoe). So to hell with them, I feel really sorry for such people. If moreover, by the same residue-and-carry method also Goldbach's conjecture (each even number beyone 4 is the sum of two prime) can be proven directly (in 10 pages) and this is also ignored, would you then not become cynical about math-professionals?

Best regards, NFB.
(Good news last week: my book "Associative Digital Network Theory" is accepted by a wellknown publisher for publication .. finally it may occur before my 70-th birthday

Hi ITIWBS,

Quote from a science historian:

"Mathematics is not a careful march down a well-cleared highway,
but a journey into a strange wilderness where explorers often get lost.
-- Rigour is a sure sign to the historian that the maps have been made,
and the real explorers have gone elsewhere."
(W.S.Anglin, science historian). . . Is'nt that a beauty ?

For more, see http://home.vianetworks.nl/users/benschop/search.htm

NotaFBene, sorry to be so long getting back to you. Personal affirs in turmoil at the moment and next week probably tied up with packing for an upcoming move.... ...then I may have a month or two without events driving me (I hope).

NotaFBene, the ITIWBS acronym, for the first thought that came to my mind, in the spirit of the dysthymic robot from the h2g2 movie, when I found that my first choice for a username had already been taken:

I

T hought

I t

W ould

B e

S o



At any rate, I'm home again after almost a years absence, much to my amazement found the vehicle still driveable, and have a host of things to do cleaning up around the house after a almost a year of unchecked weathering.

You should try joining some of the conversational threads. (To be found in any of the conversational forums screens in the upper and lower right hand corners.)

I'll be looking for your book when I'm out shopping.

Hi ITIWBS,

I thought so too
And welcome to your own home again.
As for looking at other conversations: I do, once in a while, and
- as some people told me here - I understand I'm in the wrong place ;-(
All the best, NFB

You might try this one: http://www.bbc.co.uk/dna/h2g2/brunel/F2606954?thread=5876706&skip=20&show=20 BBC - h2g2 - A Conversation Forum Back to the 'grind', as they say.

NotaFBenene, some other threads you might try. http://www.bbc.co.uk/dna/h2g2/brunel/A530560 BBC - h2g2 - The Myth Of 42 [(5-3+0+5) * (6+0) = 42] http://www.bbc.co.uk/dna/h2g2/brunel/F64903?thread=844950 BBC - h2g2 - A Conversation Forum http://www.bbc.co.uk/dna/h2g2/brunel/A22174526 BBC - h2g2 - The Quite Interesting Society

My new book "Associative Digital Network Theory" just was released by Springer Verlag (may 2009): http://www.springer.com/computer/communications/book/978-1-4020-9828-4

Among others, it contains short and direct proofs of FLT and Goldbach's conjecture (Chapters 8 and 9), along the lines given here.
That is: FLT by starting with FST (Fermat Small Theorem) using mod p^k arithmetic and analysis of the group of units (semigroup theory).
As it were a 'residue-and-carry' method (do Goldbach mod m_k where m_k = \prod. first k primes).