Re: Fermat's Lost Proof

tranders@xxxxxxxxxxx wrote:
>
> This document describes an interesting conjecture regarding
> what is known as "Fermat's Last Theorem". The theorem roughly
> states:
>
> Given the equation a^n x b^n = c^n there are no (positive)
> whole number solutions for n>2

It's conventional to denote addition by a plus sign, i.e. "+".
That key is just to the left of the backspace on your keyboard.

I presume you are aware that the terms on the left hand side
are added not multiplied! If the left side is a product then
the equation has trivial solutions for any integer n, i.e.
(for non-zero n) c = +/- a * b for any integer pair a, b
(in which the -ve option can be chosen when n is even).

(Here I've used the conventional symbol "*" for a product;
this is located above the "8" on your keyboard.)

> [...]
>
> Surely an elegant proof would not have relied on convoluted
> journeys through arcane regions of number theory.

Who says it was elegant, even if it existed? I think Fermat
in his marginal note claimed it was "wonderful", which may
not mean the same thing (although it's not an unreasonable
assumption in the circs).

Also, in maths as in many walks of life simple questions
don't always have simple answers. Logic theorists have
devised theorems whose shortest proof is demonstrably
inordinately long, apparently out of all proportion to
the "simplicity" of the theorem statement, and even if
these may be slightly "artificial" they illustrate the
point.

> I also wondered why present day mathematicians have
> claimed that Fermat did not have a proof at all

The book he was reading, Bachet's edition of Diophantus,
and in which he wrote his marginal note, was what first
got him interested in Diophantine analysis as a young
man. In other words, his claim was made when he was
starting out.

He actually scribbled lots of marginal notes of tentative
results, and for most of these he later published proofs
("published" in the sense of challenging his pen pals
Mersenne and Wallace, etc, to solve them); but he never
again mentioned the general case (despite proving it
for n = 4 and partially for n = 3).

On that basis, most mathematicians believe he spotted
a flaw in his proof; most likely this was that unlike
rational integers, cyclotomic integers might not have
unique prime factorization, and thus that factoring
x^n + y^n as a product of linear terms (x + w^i.y)
with w a primitive p-th root of unity doesn't allow
one to conclude that each linear factor is a p-th
power of a cyclotomic integer (which if true would
allow a short proof).

> even though all his other conjectures have survived
> scrutiny.

I think he conjectured that for n a positive integer
numbers 2^2^n + 1, now called Fermat numbers, are all
prime (although it could be he just suggested they
might all be, committing himself less than he would
with a conjecture).

For n <= 4 they are prime; but for n > 4 all Fermat
numbers so far checked have turned out composite.
See http://www.prothsearch.net/fermat.html

They clearly get very large very fast as n increases.
But with some ingenuity Fermat could have factored
2^2^5 + 1 as Euler first did in the early 18th C.
So even Fermat, clever as he was, did occasionally
miss a trick!

> looking through the various references in the book,
> I was especially drawn to Euclid's triples and decided
> to see if there were any similar patterns with higher
> powers.
>
> [...]

Those arrays of numbers you constructed are well-known
in the study of "Finite Differences".

I very much doubt you'll find any useful connection
between them and FLT. But you never know, something
interesting may turn up, so good luck anyway!


Cheers

John R Ramsden (jhnrmsdn@xxxxxxxxxxxx)

* Drop m from com to reply
* "From" address is defunct

.

Relevant Pages

  • Re: Some perspective, math society
    ... It is incredible that the math community is not looking for a short ... make a very powerful argument for the existence of a short proof. ... gives creedence to the conjecture "that Fermat could not or did not ... In order to prove this conjecture one must ...
    (sci.skeptic)
  • Re: Fermats Last Theorem simple proof impossible?
    ... mike3 wrote: ... Fermat's Last Theorem using only the mathematics Fermat would have had ... Fermat was trying to re-ignite interest in number theory. ... The latter is, of course, conjecture. ...
    (sci.math)
  • Re: Simple proof of FLT: Why is it impossible?
    ... like Fermat claimed to have for his famous theorem, ... It is known that descent arguments (such as the Fermat argument ... to prove FLT and perhaps even the Beal Conjecture? ... Langlands Conjectures turn out to be head-slappingly obvious ...
    (sci.math)
  • Re: JH is just making fools out of you
    ... >> conjecture based on a handful of observations, ... look at James Harris' thread on his twin ... in which Fermat gave or knew ... correct, and when Fermat gave an incorrect conjecture, ...
    (sci.math)
  • Re: A QUESTION ABOUT IRREDUCIBLE POLYNOMIALS
    ... Pis either irreducible or a product of two irreducible polynomials whose degrees equal that of P. ... By Gauss' lemma, any factorization of f ... in Qcan be scaled by constant rational multiples so as to yield a ... This proves Conjecture. ...
    (sci.math)