From: Osher Doctorow, Ph.D. [log in to unmask], Mon., Sept. 25, 2000,
10:03PM
Hello, hello. Thank you for waking me up. This reminds me of a curious
matter/mystery. As we all recall, Fermat numbers are of form 2^^m + 1 and
the Fermat primes are among those Fermat numbers for which m = 2^^k for some
positive integer k. The Mersenne numbers are like Fermat numbers but with +
replaced by - and m prime, and the Mersenne primes are among the Mersenne
numbers but are not all the Mersenne numbers. So far so good. These facts
are elementary. Next consider the elementary fact that 2^^m - 1 = 2^^m -
1^^m = (2 - 1)P(2) where P(2) is a polynomial of degree m-1 in 2, that is,
2^^(m-1) + 2(m-2) + ... 1^^(m-1). Look at the factor (2 - 1) above, which
seems trivial, since it is 1. However, in 2^^m - 1, if 1 is replaced by
b^^m for some positive integer b and 2^^m is replaced by a^^m for some
positive integer a, then (a - b) is a factor of a^^m - b^^m.
Fermat stated, concerning his Last Theorem, that he had a wonderful proof
which was too big to fit into the margins of his notebook, more or less.
Look at the word "margin" and tell me what you see. Imagine that you have
just awakened from sleep by somebody saying "Hello?" The person then says:
"Margin." In either French or English, you think "difference" more
frequently than you think "margin of book". There might also be another
factor or even a limit involved, but the idea of difference is central to
the use of "margin" in mathematics and even economics (one of the very,
very few cases in which I cite anything from economics). I am going to
conclude this message, before returning to sleep (it is 10:25PM here), by
asking whether or not Fermat's claimed proof involved the "margin" as the
factor a - b of the difference a^^m - c^^m of the equation b^^m + c^^m =
a^^m for a, b, c positive integers (m > 2). If you conclude that this was
what Fermat meant, then I will tell you the next part of the story next time
(which time will be among those times when I am awake, of course). If not,
then we may have to let the story sleep a bit longer. It does not matter.
Time when one is asleep passes very rapidly ordinarily. That reminds me of
another curious fact - but here I bid you Good Night.
Osher
----- Original Message -----
From: <[log in to unmask]>
To: <[log in to unmask]>
Sent: Monday, September 25, 2000 5:39 PM
Subject: Mersenne: Hello?
> Oh, Hi,.., so there is still someone here!
> For a week there I thought the world had ended. I haven't recieved a
> mersenne email for a week now. ....
> Dan
> Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
> Mersenne Prime FAQ -- http://www.exu.ilstu.edu/mersenne/faq-mers.txt
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