>One of the curious points about the Bogdanovs is that they seem to
>be impervious to collegial correction, but consider the case of
>Fermat's (?) last theorem scrawled in the margin of some book that,
>excuse the lack of superscripts,
> x to the nth + y to the nth = z to the nth
> cannot be true for any value of N higher than 3
>Can you think of anything less rigorous than a bare theorem scrawled
>in a margin. Maybe in a hundred years (????) someone will prove all
>their unproved assertions.
In defense of Fermat, he stated that the margin was insufficient to contain the proof! (And he stated that the proposition was true for n greater than 2.)
This theorem seems to have a better fate than Fermat's cobjecture that every number fo the form 2^(2^n) + 1 was prime: it seems to be true only for n<=4. Euler was the first to show that the 5th Fermat number was composite. (Even this flawed conjecture bore some fruit: Gauss showed that one copuld only construct a regular polygon with straightedge and compass if it had a number of sides equal to the product of a power of 2 and one or more distinct Fermat primes.)
You are certainly correct that Fermat did not prove many of the things that he asserted (including his so-called little theorem -- is p is a prime and a is relatively prime to p, then a^(p-1) - 1 is divisible by p), but his intuition about what was true was outstanding. That it took Euler to prove many of Fermat's most insightful "theorems" should certainly not detract from his work.
In further defense, Fermat did not publish most of his mathematical work--most of his extant work comes from letters he sent that others kept. And, unlike the Bogdanovs, Fermat was proposing unproven, but ostensibly true things about mathematics. W.W. Rouse Ball, in A Short Account of the History of Mathematics, mentions that Fermat must be seen as the co-founder, with Pascal, of probability theory.
--tim francis-wright (whose favorite integer, 17, is a Fermat prime)