From: Osher Doctorow, Ph.D. [EMAIL PROTECTED], Mon. Sept. 18, 2000, 6:06PM
In attempting to find a shorter proof of Fermat's Last Theorem (FLT) than
the current one, I have become interested in the rather curious expressions
g(x,y) = 1 - x + y and f(x,y) = x + y - xy. I have elsewhere indicated how
an n-dimensional generalization of the real conjugate of one or both of
these expressions would lead to a super-short proof of FLT - the real
conjugate of g(x.y) is defined as 1 + x - y and the real conjugate of f(x,y)
is defined as x + y -xy. Abstracts of 46 of my over 100 papers can be found
on the internet at the Institute for Logic of the University of Vienna,
http://www.logic.univie.ac.at, select ABSTRACT and then BY AUTHOR and then
my name. However, I am here presenting a slightly different problem which
is indirectly linked to Mersenne Primes via the Sophie Germain Prime - FLT -
Fermat Number - Mersenne Number linkage. To shorten the presentation, I
became curious about the expression (x - y)^^2 = x^^2 - 2xy + y^^2 and how
it relates to f(x,y) = x + y - xy. Is it possible that either f(x,y) or
f1(x,y) = x + y - 2xy is actually representable as a power of x - y? The
answer seems to be no in the ordinary sense of power or exponent, but what
about ordinals, cardinals, etc.? Is it possible that below the second
power, there is a different infinite continuum of exponents from the
familiar one? If so, then 2 would be the boundary between two continua, and
this might explain its importance in FLT itself. Setting f(x,y) = xy yields
x + y = 2xy or x + y - 2xy = 0, or f1(x,y) = 0. Anticipating things, let
us write "a" as the exponent of x-y involved in f1(x,y), so (x-y)^^a = 0 is
equivalent to f1(x,y) = 0. However, I have shown elsewhere that xy is a
"different animal" from x or y not just in the sense of a polynomial but in
a sense analogous to the quaternion/octonion multiplication of
vectors/pseudovectors u and v, which is written uv. Thus, f(x,y) = xy has
interest in itself as a sort of extreme/semi-extreme case, which means that
(x-y)^^a is of interest in itself. We also have the equations x + y + xy =
xy, which is equivalent to x + y = 0, and x + y + xy = -xy, which says that
x + 2xy + y = 0, which seems to give rise to (x+y)^^a, and x - y - 2xy
= -2xy which says x - y = 0. Notice that x - y - 2xy is obtained from x +
y + 2xy by replacing y by -y. Finally, the perfect numbers are sums of
their smaller divisors (among integers), which means that xy/x + xy/y +
other expressions equal the perfect number xy if xy is the product x times
y. Also, any even perfect number has form 2^^(r-1)(2^^r - 2) if 2^^r and
r are prime, from Euclid's Elements. As far as I know, no odd perfect
number exists. Thus, some sort of xy factorization could be conjectured for
all perfects (or at least for very many). The Mersenne primes have form
2^^r - 2 for r prime, of course. I hope to continue this soon.
Osher Doctorow
_________________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
Mersenne Prime FAQ -- http://www.exu.ilstu.edu/mersenne/faq-mers.txt
