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Topic: From Fermat little theorem to Fermat Last Theorem
Replies: 62   Last Post: Mar 14, 2013 9:59 PM

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 Brian Q. Hutchings Posts: 6,427 Registered: 12/6/04
Re: From Fermat little theorem to Fermat Last Theorem
Posted: Mar 7, 2013 11:29 PM

Munk said that this must have been Fermat's proof, and
I can only verify it up to the very last page
of the alleged proof, which may be poorly stated
due to "English as a second language," although
it is mostly very elementary.
Munk was well-qaulified as a student of Prandtl in aerodynamics,
as far as mathematical physics goes, at Goettingen U.;
he was one of the pioneers at NACA, the predecessor to NASA,
albei mainly with planar slices of airfoils.

The book is actually quite amuzing, and he states
in the chapters after the main result, that most mathematicians
would think of his "congruence surds" as "p-adic numbers,
which they are not." Yes, the p-adics are a big key
to this, as they are in quantum mechanics, but it is really
just a matter of "adding-machine mechanics," which a few
of the classical math-folks had actually developed,
before anyone else: Pascal, Liebniz and Fermat.

Although the congruence surds have the same problem
as p-adics, namely that they have no simple, "archimedean
valuation" or absolute scalar value, that is not important
to this theorem, which merely proves the impossiblity
of certain congruences, teh ones that are akin
to the Pythagorean theorem, using exponents other than two.

Here, I will just supply a note as to the hearsay about Fermat,
not having been able to prove this "miraculous" result,
as he wrote in the margin of his _Bachet's Diophantus_ book
of old number problems. Well, he made no other known errors,
and this includes the sole remaining unsloved problem
of Fermat, the characterization of the Fermat numbers
(of the form, 2^(2^h) + 1, h = 0, 1 etc. Well,
he had merely congectured that they were all prime, and
he later retracted this ideal in a letter to Bernard
de Frenicle; so, theresville.

Furthermore, this may have been one of his earliest proofs
in the theory of numbers -- which science he created,
mostly using right trigona -- and this is highlighted
by the fact taht he *later* produced the proof
for teh very special case of the exponent, n = 4. All
of the other cases can be made for only prime exponents,
since it is easy to prove that composite exponents
-- other than 2x2 -- reduce to the case for prime ones.

--sincerely, R. Brian Hutchings

Date Subject Author
11/27/12 John Jens
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