From Osher Doctorow
COPYRIGHT NOTICE
Fermat's Last Theorem - Did It Discovery Feedback?
Copyright By Owner Osher Doctorow Ph.D.
First Published 2006
Fermat's Last Theorem, discovered by Pierre De Fermat of 1600s France
who came before Sir Isaac Newton and Leibniz but whose life overlapped
with Newton's and Leibniz' for Fermat's last two decades or so, says:
1) x^n + y^n = z^n has no positive integer solutions x, y, z for
integer n > 2
Fermat codiscovered probability with Pascal, discovered modern number
theory which led to cryptography, discovered analytic/Cartesian
geometry before Descartes, discovered several calculus equations before
Sir Issac Newton and Leibniz, upstaged Descartes in optics by
predicting the slowdown of light in water, etc.
The proof of (1) by Andrew Wiles of Princeton is so difficult that it
has led to speculation in number theory that it may not be as
fundamental a property as had been thought, since so many fundamental
theoretical properties have easier or even multiple proofs.
However, look at this:
2) x^n - z^n = (x - z)(x^(n-1) + x^(n-2)z + x^(n-3)z^2 + ... +
x^2z^(n-3) + xz^(n-2) + z^(n-1))
which is well known in algebra. The right hand side is symmetric in
the second factor or "balanced", and is reminiscent of the feedback
equations discussed in the last few sections of my
Nonrenormalization-Renormalization thread.
What does this mean? I would conjecture or guess that Fermat, who by
the way was a Magistrate in France by profession and only a
mathematical amateur and who had a profile very similar to Sir Isaac
Newton, was actually part of a French-U.K. rivalry to research feedback
and other engineering concepts. His optics light speed ideas, for
example, may have been inspired by the British Admiralty's engineering
work on submarines in the late 1500s and the Dutch experiments on a
first submarine in the Thames around 1620. Fermat's Last Theorem was
arguably not intended as a purely number theoretic theorem, or even as
a "negative" ("there does not exist") theorem, but was exploratory into
symmetric properties of feedback. His "wonderful proof in the margin",
which was never found in his notes, may refer to the idea of a margin
in both English and French as reflecting a difference (for example, the
difference between a written or printed text portion and the size of
the whole page, but here the difference (2)).
It might be argued that other algebraic equations involve symmetry or
"balance", including (x + y)^n, (x - y)^n, etc. Some of them may be
"coincidental", although it should be noted that both x^n - z^n and x -
y are differences, and that PI P(A-->B) = 1 + y - x with y = P(AB) and
x = P(A) is fundamentally a difference. Whether Fermat discovered
Probable Influence/Causation (PI) and kept it secret is unknown, but
both Fermat and Sir Isaac were extremely secretive. Fermat would not
even reveal his methods in public usually but only his results, which
so irritated Descartes (a "professional" mathematician) that the latter
spent his life trying to oust Fermat from his Magistrate position after
Fermat humiliated him twice and after Descartes was forced to admit his
errors in public. Readers should look up the various internet
histories of Fermat, e.g., University of St. Andrews under "Fermat
biography" keyword, but also books in public libraries including older
ones.
Osher Doctorow
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