| Date: Tuesday, September 15, 2009
Title: Smooth solutions to the ABC Equation
Abstract: The ABC equation is the linear Diophantine equation
A+B+ C=0, where one considers solutions (A, B, C) that are
relatively prime integers. The height
of a solution is H= max (|A|, |B|, |C|). The
famous ABC conjecture studies the size of the
conductor R, which is the product of all primes dividing ABC
(taken without multiplicity), and says that to get infinitely many solutions,
R must be as large as H^c for some positive c.
Here we study instead solutions (A, B, C) having
only small prime divisors. We define the smoothness of a solution as
S = max{p: p a prime dividing ABC}. How small can
S be as a function of H so that there are still infinitely
many solutions? We determine--assuming unproved hypotheses--
what the right order of magnitude of S should be: it is
S= (log H)^{c} for some positive constant c.
This is joint work with K. Soundararajan.
Speaker: Jeffrey Lagarias
Institution: Univ of Michigan
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