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Mathematics Colloquium


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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