goldberg@decwrl.UUCP (David Goldberg) (07/20/83)
The Mordell conjecture says that if you have a polynonmial in two variables
with rational coefficients (like x^n - y^n - 1) and if when you
think of it as a Riemann surface it has genus >= 2, then it has finitely
many rational solutions.
If you take the Fermat equation x^n + y^n = z^n, and divide by z^n, you
get the equation t^n + s^n - 1 = 0, where t = (x/z) and s = (y/z).
When n > 2, its genus is >=2, so the Mordell conjecture implies that
for each n > 2, Fermat's equation has at most finitely many solutions.
Faltings has circulated a preprint of his proof of the Mordell conjecture,
but it hasn't been independently checked yet.
david goldberg
{decvax, ucbvax}!decwrl!goldberg